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Practical Treatise 



ON THE 



Differential and Integral 

Calculus, 

WITH SOME OF ITS 

APPLICATIONS TO MECHANICS AND 
ASTRONOMV. 



ST 

WILLIAM GV PECK, Ph.D., LL.D., 

Profetfior of Mathematics and Astronomy in Columbia CoUem 
and of Mechanics in th* School of Mines. 




NEW YORK • : • CINCINNATI • ! • CHICAGO 

AMERICAN BOOK COMPANY 



PUBLISHERS' NOTICE. 

PECK'S MATHEMATICAL SERIES 

CONCISE, CONSECUTIVE, AND COMPLETE. 



I. FIRST LESSONS IN NUMBERS. 
II. MANUAL OF PRACTICAL ARITHMETIC. 

III. COMPLETE ARITHMETIC. 

IV. MANUAL OF ALGEBRA. 
V. MANUAL OF GEOMETRY. 

VI. TREATISE ON ANALYTICAL GEOMETRY. 
VII. DIFFERENTIAL AND INTEGRAL CALCULUS, 
fill. ELEMENTARY MECHANICS (without the Calculus). 

IX. ELEMENTS OF MECHANICS (with the Calculus). 



Note. — Teachers and others, discovering errors in any of 
the above works, will confer a favor by communicating them 
to us. 



9*i 




AUG 2 



*>\P Copyright, 1870, by William G. Peck. 

Copyright, 1898. by Guy D. Peck. 



peck s CAL. 



2n 

1896. 



• fitCElVED. 






•v-U 1T 



PREFACE 



The following pages are designed to embody the course 
of " Calculus and its Applications," as now taught in 
Columbia College. This course being purely optional, ia 
selected by those only who have a taste for mathematical 
studies, and it is pursued by them more with reference to 
its utility than as a means of mental discipline. These 
circumstances have given to the present work a practical 
character that can hardly fail to commend it to those who 
study the Calculus for the advantages it gives them in 
the solution of scientific problems. To meet the wants 
of this class of students, great care has been taken to avoid 
superfluous matter; the definitions have been revised and 
abbreviated; the demonstrations have been simplified and 
condensed , the rules and principles have been illustrated 
and enforced by numerous examples, so chosen as to famil- 
iarize the student with the use of radical and transcen- 
dental quantities; and finally, the manner of applying the 
Calculus has been exemplified by the solution of a variety 
of problems in Mechanics tend Astronomy. 

The method employed in developing the principles of 
the science is essentially that, of .Leibnitz. This method. 



4: PREFACE. 

generally known as the method of infinitesimals, has been 
adopted for several reasons: first, it is the method adopted 
in all practical investigations; second, it is the method 
most easily explained and most readily comprehended ; 
and third, it is a method, as will be shown in the final 
note, identical in results with the more commonly adopted 
method of limits, differing from it chiefly in its phraseology 
and in the simplicity of its results. 

The author cannot conclude this prefatory note with- 
out acknowledging his obligations to those students who, 
from year to year, have been willing to turn aside from 
the attractive pursuit of classical learning to engage in the 
sterner study of those processes that have contributed so 
much to the progress of modern science. Without their 
interest and co-operation this book would never have been 
written. He would also take this opportunity to express 
his thanks to his distinguished colleague, Professor J. H. 
Van Amringe, not only for many valuable suggestions 
made during the progress of the work, but also for much 
effective labor in reading and correcting the proofs as they 
oame from the press. 

Columbia College, 

tfnv<Ymhe,r 2Ath. 1*70. 



CONTENTS. 



PART 1.— DIFFERENTIAL CALCULUS. 

I. Definitions and Introductory Remarks. 
krt. PAei 

1. Classification of Quantities 9 

2. Functions of one or more Variables 9 

3. Geometrical Representation of a Function 10 

4. Differentials and Differentiation 10 

5. Geometrical Illustration 12 

6. Infinites and Infinitesimals 13 

7. General Method of Differentiation 14 

II. Differentiation of Algebraic Functions. 

8. Definition of an Algebraic Function 15 

9. Differential of a Polynomial 15 

10. Differential of a Product 16 

11. Differential of a Fraction 17 

12. Differential of a Power 18 

13. Differential of a Radical 18 

III. Differentiation of Transcendental Functions. 

14. Definition of a Transcendental Function 24 

15. Differential of a Logarithm 25 

16. Differential of an Exponential Function 26 

17. Differentials of Circular Functions 29 

18. Differentials of Inverse Circular Functions 32 

IV. Successive Differentiation and Development of 
Functions. 

19. Successive Differentials 35 

20. Successive Differential Coefficients 35 

21. McLaurin's Formula 37 

22. Taylor's Formula , 41 



b CONTENTS. 

V. Differentiation of Functions of two Variables and op 
Implicit Functions. 

An. PAGE 

23. Geometrical Representation of a Function of two Variables 45 

24. Differential of a Function of two Variables 46 

25. Notation for Partial Differentials 47 

26. Successive Differentiation of Functions of two Variables . . 48 

27. Extension to three or more Variables 5(1 

28. Definition of Explicit and Implicit Functions 50 

29. Differentials of Implicit Functions 50 

PART II— APPLICATIONS OF THE DIFFERENTIAL 
CALCULUS. 

I. Tangents and Asymptotes. 

30. Geometrical Representation of first Differential Coefficient. . 53 

31. Applications 54 

32. Equations of the Tangent and Normal 55 

33. Asymptotes 57 

34. Order of Contact 50 

II. Curvature. 

35. Direction of Curvature GO 

36. Amount of Curvature 62 

67. Oscillatory Circle 63 

38. Radius of Curvature 64 

39. Co-ordinates of Centre of Curvature 65 

40. Locus of the Centre of Curvature 66 

41. Equation of the Evolute of a Curve 67 

III. Singular Points of Curves. 

42. Definition of a Singular Point (p* 

43. Points of Inflexion fly 

44. Cusps 7(i 

45. Multiple Points 71 

46. Conjugate Points 7e 

IV. Maxima and Minima. 

47. Definitions of Maximum and Minimum 74 

48. Analytical Characteristics 75 

19. Maxima and Minima of a Functiun of two Variables . 8f 



CONTENT.^. 7 

V. Singular Values op Function s. 

Art. PIGH 

50. Definition and Method of Evaluation 88 

VI. Elements of Geometrical Magnitudes. 

51. Differentials of Lines, Surfaces, and Volumes 91 

VII. Application to Polar Co-ordinates. 

52. General Notions, and Definitions 93 

53. Useful Formulas 94 

54. Spirals 97 

55. Spiral of Archimedes 98 

VIII. Transcendental Curves. 

56. Definition 99 

57. The Cycloid 99 

58. The Logarithmic Curve 101 

PART III.— INTEGRAL CALCULUS. 

59. Object of the Integral Calculus 102 

60. Nature of an Integral 102 

61. Methods of Integration ; Simplifications 104 

62. Fundamental Formulas 104 

63. Integration by Parts 114 

64. Additional Formulas 115 

65. Rational and Entire Differentials 120 

66. Rational Fractions 121 

67. Integration by Substitution, and Rationalization 128 

68. When the only Irrational Parts are Monomial 128 

69. Binomial Differentials 130 

70. Integration b}' Successive Reduction 133 

71. Certain Trinomial Differentials 141 

72. Integration by Series 144 

73. Integration of Transcendental Differentials 145 

74. Logarithmic Differentials 145 

75. Exponential Differentials 147 

76. Circular Differentials 148 

77. Integration of Differential Functions of two Variables 153 

PART IV.— APPLICATIONS OF THE INTEGRAL CAL- 
CULUS. 
I. Lengths of Plane Curves. 

78. Rectification 156 



* 



o OONTENTS. 

II. Areas of Plane Curves. 

A*t. FA8I 

79. Quadrature 161 

III. Areas of Surfaces of Revolution. 

SO Surfaces Generated by the Revolution of Plane Curves. ... 185 

IV. Volumes of Solids of Revolution. 

81. Cubature 167 



PART V.— APPLICATIONS TO MECHANICS AND 
ASTRONOMY. 

I. Centre of Gravity. 

82. Principles Employed 171 

83. Centre of Gravity of a Circular Arc 173 

84. Centre of Gravity of a Parabolic Area 174 

85. Centre of Gravity of a Semi-Ellipsoid of Revolution 175 

86. Centre of Gravity of a Cone 176 

87. Centre of Gravity of a Paraboloid of Revolution 176 

II. Moment of Inertia. 

88. Definitions and Preliminary Principles 176 

89. Moment of Inertia of a Straight Line 177 

90. Moment of Inertia of a Circle 178 

91. Moment of Inertia of a Cylinder 179 

92. Moment of Inertia of a Sphere 180 

III. Motion of a Material Point. 

93. General Formulas 181 

94. Uniformly Varied Motion 182 

95. Bodies Falling under influence of Constant Force 183 

96. Bodies Falling under action of Variable Force 186 

97. Vibration of a Particle of an Elastic Medium 188 

98. Curvilinear Motion of a Point 190 

99. Velocity of a Point rolling down a Curve 192 

100. The Simple Pendulum 193 

101. Attraction of Homogeneous Spheres 195 

102. Oibital Motion 199 

103. Law of Force 204 

104. Note on the Methods of the Calculus 20$ 



PART I. 

DIFFERENTIAL CALCULUS. 



I. Definitions and Introductory Remarks. 

Classification of Quantities. 

1. The quantities considered in Calculus are of two 
kinds: constants, which retain a fixed value throughout 
the same discussion, and variables, which admit of all pos- 
sible values that will satisfy the equations into which they 
enter. The former are usually denoted by leading letters 
of the alphabet, as a, b, c, etc.; and the latter by final 
letters, as x, y, z, etc. ; particular values of variable quan- 
tities are denoted by writing them with one or more dashes, 
us x', y", z'", etc. 

Functions of one or more Variables. 

2. Relations between variables are expressed by equa- 
tions. In an equation between two variables, values may 
be assigned to one at pleasure; the resulting equation de- 
termines the corresponding values of the other. The one 
to which arbitrary values are assigned is called the inde- 
pendent variable, and the remaining one is said to be a 
function of the former. If an equation contain more than 
two variables, all but one are independent, and that one is 
I function of all the others. The fact that a quantity 

1 # 



10 DIFFERENTIAL CALCULUS. 

depends on one or more variables may be expressed at 
follows : 

y =/0*0 ; « = <p(», y) ; ^fe y, «) = o. 

The ,/zrs^ shows that y is a function of x, the second that 
z is a function of x and y, and the third that x, y, and z, 
depend on each other, without pointing out which is a 
function of the other two. 



Geometrical representation of a Function. 

3. Every function of one variable may be represented b\ 
the ordinate of a curve, of which the variable is the cor- 
responding abscissa. For, let y be a function of x, and 
suppose x to increase by insensible gradations from — oo 
to + oo. For each value of x there will be one or more 
values of y, and these, if real, will determine the position 
of a point with respect to two rectangular axes. These 
points make up a curve, at every point of which the rela 
tion between the ordinate and abscissa is the same as that 
between the function and independent variable. This 
curve is called the curve of the function. 

For values of x that give imaginary values of y there are 
no points; for those that give more than one real value of 
y there is a corresponding number of points. 

In a similar manner it may be shown that a function 
of two variables represents the ordinate of a surface of 
which the variables are corresponding abscissas. 

Differentials and Differentiation. 

4. Of two quantities, that is the less whose value is 
nearer to — oo, and that is the greater whose value is 
nearer to 4- x . A quantity is said to increase when it 



DEFINITIONS AND INTRODUCTORY REMARKS. 11 

approaches + oo , and to decrease when it approaches — c/d 
The ordinates of a curve originate from the axis of X and 
of any two, that is the greater whose extremity is nearer 
4- x> , and that is the less whose extremity is nearer — oo ; 
in like manner of two abscissas, that is the greater whose 
extremity is nearer + oo , and that the less whose extremity 
is nearer — oo . 

In what follows we shall suppose the independent varia- 
ble to increase by the continued addition of a constant but 
infinitely small increment. For every change in the value 
of the variable there is a corresponding change in the valup 
of the function. In some cases, as the variable increases, 
the function increases; it is then said to be an increasing 
function : in other cases the function decreases as the 
variable increases ; it is then said to be a decreasing func- 
tion. In all cases, the change in value is called an incre- 
ment ; for increasing functions the increment is positive, 
and for decreasing functions it is negative. The increment 
of the function is always infinitely small, but it is not con- 
stant, except in particular cases. 

The infinitely small increment of the independent varia- 
ble is called the differential of the variable, and the cor 
responding increment of the function is called the differ- 
ential of the function. Hence, the differential of a quantity 
is the difference between two_ consecutive values of that 
quantity. It is to be observed that the difference is always 
found by taking the first value from the second 

The operation of finding a differential is called differen 
tiation. The object of the differential calculus is to ex 
plain the methods of differentiating functions. 



12 



DIFFERENTIAL CALCULUS. 




Fig. 1. 



Geometrical Illustration. 

5. Let KL be a curve in the plane of the rectangulai 
axes OX and OY, and let OA and OB be two abscissas 
aiffering from each other by an infi- 
nitely small quantity AB. Through 
A and B draw ordinates to the 
curve, and let PR be parallel to 
OX. OA and OB are consecutive 
abscissas, AP and BQ are consecu- 
tive ordinates, and P and Q are 
consecutive points of the curve. 
The part of the curve PQ does 
not differ sensibly from a straight line, and if it be pro 
longed toward T, the line PT is tangent to the curve at 
P. If we denote any abscissa OA by x, and the corre- 
sponding ordinate by y, we have, 

y =/(*)• 

Th<» line AB is th> differential of the independent varia- 
ble, denoted by the symbol dx; RQ is the differential of 
the function, denoted by dy; and PQ is the differential 
of the curve KL, denoted by ds. 

The right angled triangle, RPQ, gives the relation 
ds -.- Vdx 7 + dy\ Denoting the angle RPQ by 6, wt 
have, from trigonometry, 



tand 



dx 



sin0 



dy 

ds 



dy 



vdx* -f dy* 



dx 
ds 



dx 



*nd, cosd = -j- = — 



Vdx* +■ dy* 



DEFINITIONS AND INTRODUCTORY REMARKS. 13 



Infinites and Infinitesimals. 

6. A quantity is infinitely great with respect to another 
when the quotient of the former by the latter is greater 
than any assignable number, and infinitely small with re- 
spect to it, when the quotient is less than any assignable 
number. If the term of comparison \s finite, quantities of 
the former class are called infinites, and those of the latter 
infinitesimals. 

Infinites and infinitesimals are of different orders. Let 
us assume the series. 



a a a „ s 

-n, -r, — , a, ax, ax , ax . 



In which a is a finite constant and x variable. If we 
suppose x to increase, the terms preceding a will diminish, 
and those following it will increase ; when x becomes greater 

than any assignable quantity, — becomes infinitely small 

with respect to a, and because each term bears the same 
relation to the one that follows it, every term in the series 
is infinitely small with respect to the following one, and 
infinitely great with respect to the preceding one. The 
quantity ax being infinitely great with respect to a finite 
quantity, is called an infinite of the first order; ax 2 , ax 3 , 
etc., are infinites of the second, third, etc., orders. The 

quantity - being infinitely small with respect to a finite 

quantity is called an infinitesimal of the first order; -j, -j, 

x x 

etc, are infinitesimals of the second, third, etc., orders. 

It is to be observed that the product of two infinitesi- 
mals of the first order, is an infinitesimal of the second 



t4 DIFFERENTIAL TALCULUS. 

order. For, let x and y be infinitely small with respect 
to 1, we shall have, 

1 : x : : y : xy. 

Hence, xy bears the same relation to y that x does to 1, 
that is, it is infinitely small with respect to an infinitesimal 
of the first order; it is therefore an infinitesimal of the 
second order. The product of three infinitesimals of the 
first order is an infinitesimal of the third order, and so on. 
[n general, the product of an infinitesimal of the m th order 
by one of the n th order, is an infinitesimal of the (m + w) tb 
order. The product of a finite quantity by an infinitesimal 
of the n th order* is an infinitesimal of the n th order. 

From the nature of an infinite quantity, its value will 
not be sensibly changed by the addition or subtraction of 
a finite quantity. A finite quantity may therefore be dis- 
regarded in comparison with an infinite quantity. For 
a like reason an infinitesimal may be disregarded in com- 
parison with a finite quantity, or with an infinitesimal of a 
lower order. Hence, whenever an infinitesimal is con- 
nected, by the sign of addition, or subtraction, with a finite 
quantity, or with an infinitesimal of a lower order, it may 
be suppressed without affecting the value of the expression 
into which it enters. 

General method of Differentiation. 

7. In order to find the differential of a function, we give 
to the independent variable its infinitely small increment, 
and find the corresponding value of the function; from 
this we subtract the preceding value and reduce the result 
to its simplest form; we then suppress all infinitesimals 
which are added to, or subtracted from, those of a lowej 
order, and the result is the differential required. 



DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 15 

This method of proceeding is too long for general use, 
and is only employed in deducing rules for differentiation 

II. Differentiation of Algebraic Functions. 

Definition of an Algebraic Function. 

8. An algebraic function is one in which the relation 
between the function and its variable can be expressed by 
the ordinary operations of algebra, that is, addition, sub- 
traction, multiplication, division, formation of poioers de- 
noted by constant exponents, and extraction of roots indicated 
by constant indices. Thus, 

y* — 2px, y = ax 2 — Vbx, and Vy = ^a' 2 x — bx*. 

are algebraic functions. 

Differential of a Polynomial. 

9. Let a and c be constants, and r, s, t, functions of x\ 

assume 

y = ar + s — t 4- c ; ( 1 ) 

iu which y denotes the polynomial in the second member, 
and is therefore a function of x. 

If we give to x the increment dx, the functions y, r, s, 
and /, receive corresponding increments dy, dr, ds, dt, and 
we have, 

y -f dy = a(r + dr) + (5 + ds) - (t + dt) 4- c (2) 

subtracting (1) from (2), we have, 

dy = adr + ds — dt (3) 

Hence, to differentiate a polynomial, differentiate each 
term separately and take the algebraic sum of the results. 

Comparing (1) and (3) we see,f'rsf, that a constant factor 
remains unchanged, and secondly, that a constant term 
disappears by differentiation. 



16 DIFFERENTIAL CALCULUS. 

Differential of a Product. 

10. Let r and s be functions of x. Placing their pro 
duct equal to y, we have, 

y = rs (1) 

Giving to x the increment dx, we have, as before, 

y + dy = (r + dr) (s + ds) = rs + rds 4- s^r + drds. . (2) 

Subtracting (1) from (2) and suppressing drds, which is an 
infinitesimal of the second order, we have, after replacing 
y by its value rs, 

d(rs) = rds + sdr (3) 

Hence, to differentiate the product of two functions, 
multiply each by the differential of the other, and take the 
algebraic sum of the results. 

If we suppose r = ho, we have, from the rule, 

dr ~ tdw + wdt ; 
which substituted in (3), gives, 

d(stw) = hods 4- stdio + siodt (4) 

In like manner the principle may be extended to the 
product of any number of functions. Hence, to differen- 
tiate the product of any number of functions, multiply the 
differential of each by the continued product of all the 
others, and take the algebraic sum of the results. 

Cor. — If we divide both members of (4) by stw, we have, 

d(sho) ds dt dw .„. 

-±- i — L = — + -7- + — (o) 

stw s t w 

and similarly, where there are a greater number of factors. 
Hence, the differential of a product divided by that pro- 
duct, is equal to the sum of the quotients obtained by 
dividing the differential of each factor by that factor. 



DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 17 

Differential of a Fraction. 
11. Let s and t be functions of x, and assume, 





a' 

* = ?•■■• 


■ (i) 


Giving to x the increment dx, we 


have, 






(2) 


Subtractin 


? (1) from (2), 






7 . s + ds s 

dv = - t-. — - = 


tds — sdt 



* t + dt t f + tdt' 

Replacing y by its value -, and suppressing tdt in compari- 
son with / 2 , we have, 

tds — sdt 



6)- 



(3) 



Hence, the differential of a fraction is equal to the denom- 
inator into the differential of the numerator, minus the 
numerator into the differential of the denominator, divided 
by the square of the denominator. 

If either term of the fraction be constant, its differentia] 
will be 0. When the denominator is constant, formula (3) 
becomes, 

'&)-?> « 

when the numerator is constant it becomes, 

<j)=-f » 



18 DLFFEKENTIAL CALCULUS. 

Differential of a Power. 
12. Let s be a function of x, and m a constant. Assume 

y = > m (i) 

Giving to x the increment dx. we have, 

y+dy=(s + ds) m = s m + ms™" 1 ** 



m(m — 1) , 9 . , x ,_. 

+ — r~5 6'W-2^2 + ( etc# ) (2) 



in which all the terms of the development, after the second, 
contain the square, or some higher power, of ds. Subtract- 
ing (1) from (2), we have, 

dij = ms m ~ l ds + (etc.)rZs 2 . 

Suppressing all the terms of the second member, after the 
first, in accordance with the principle laid down in Art. 6, 
and replacing y by its value, we have, 

d(s m ) = ms m - 1 ds (3) 

Hence, to differentiate any power of a function, diminish 
the exponent of the function by 1, and multiply the result H, 
the primitive exponent and the differential of the function. 



Differential of a Radical. 
IS. Let s be a function of x, and assume, 

y = Vs (i) 



DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 19 

We may write (1) in the form, 
1 

Differentiating (2), as in the last article, we have, 



1-1 1-fl 

dy = -s n ds = -s n ds (3) 

* n n K ' 



Replacing y by its value, and remembering that 
s l—n — — we have, 

d #;=!-£= (4) 



That "is, the differential of a radical of the n tL degree 
13 equal to the differ ejitial of the quantity under the 
radical sign, divided by n times the (n — l) th power of the 
radical. 

Cor. — If n = 2, we have, 

4^=^ ^ 

That is, the differential of the square root of a quantity 
is equal to the differential of the quantity under the radical 
sign, divided by tivice the radical. 

The preceding rules are sufficient to differentiate an^ 
algebraic function whatever. 



20 DIFFERENTIAL CALCULUS. 

EXAMPLES. 

1. Let y = 5x*. 

This is the product of the function x 3 by the constant 5 ; 
hence, (Art. 9), 

dy = 5d(x 3 ) =5X 3x*dx = 15x*~dz (Art. 12). 

2. Let y = x 3 + 2x 2 + 3x + 4. 

This is a polynomial. Hence, from Art. 9, 
dy = 3x 2 dx + 4^xdx + 3dx. 

3. Lety = (a + fa;) 3 . 

Considering a -f bx as a single quantity, we have, 
(Art. 12), 

dy = 3(a+bx) 2 d(a + bx) = 3 (a + bx) 2 bdx = 3b{a + ta) 2 tfar 

4. Let y = (a + fa 2 ) 71 . 

dy = n(a + frr 2 ) 71 - 1 ^ + bx 2 ) = 2te(« 4- fr^" 'W 

5. Let y = Va 2 -x 2 = (a 2 - x 2 )%. 



dy = \{a 2 - x 2 )~i X - 2xdx = 



xdx 



Va 2 —x 9 
6. Let y = (a +x) (b + 2x 2 ). 

By Art. 10, we haye, 

dy=(a + x)d(b + 2x 2 ) + {b + 2z 8 )tf(« + x) 
= (a + a)4a;c?:£ + (b + 2x*)dx ; or, rfy 
:= (6#» + 4az + £)cfa\ 



DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 21 

ax 



7. Let y 



a 2 + x 2 ' 



By Art. 11, we have, 

_ (a 2 + x 2 )d{ax) — axd(a 2 + x 2 ) _ a(a 2 -x 2 ) 
dy ~ (a 2 +x 2 ) 2 ~{a 2 +x 2 )* dX ' 



8. Let y = Vx + ^/a 2 + z*. 
dy = d[x+ (a 2 + x 2 )^ =^x + {a 2 +z2)*]~* 

X d\x+ (a 2 +x 2 )%~] 
but, d\z + (a 2 + x 2 )^~\ = dx+xdx{a 2 +x 2 )~~* 
= \x+ {a 2 + x 2 )^~](a 2 + x 2 )~~ 2 dx. 



... dv=^± y?im d z. 

2Va*+z* 
9. y = ax* — bx^ =h c. Ans. dy = (%ax 2 — \bx^)dz, 

10. y = ax~~% - bx~%± c. 

Ans. dy — — \iax~ 2 — %bx~ 2 )dx, 

11. y =(a 2 - x 2 ) 9 . Ans. dy — - ±x(a 2 - x 2 )dx. 

12. y = (2ax - x 2 ) 3 . Ans. dy = 6{a - x)(2ax - z*)'dz. 



I DIFFERENTIAL CALCLLU8. 

13. y = (a +bx n ) . Ans. dy = mnbx n ~ 1 (a + bx n ) " dx 

14. y=(2ax-\-x 2 ) n . Ans.dy=2n(a +x)(2ax + x 2 ) n ~ dx. 

15. y = x(a + x)(a 2 +<c 2 ), 

Ans. dy = (a 3 + 2a 2 x + Sax 2 + 4# 3 )flte. 

16. y= (a + a) wl (i + a;) w . 

^s. dy = (a + *r(J + »)*[ * +» "Ifc 

La + x o + xJ 

17. v =?L=f . ^rcs. dy = - , 2a N &. 

a + x {a +x) % 

(x + 4)2 ( x + 2)(x + 4) , 

18. y =^ £-, Ans. dy = v — — <& 



a + 3 * (a + 3) 



1rt z 2 -z + l . _ 2z(z-2, , 

19. y = — . Ans. dy = 7——^ Tr a dx 

y x 2 +x-l * (x 2 +x-l) 9 

dx 

20. y = Va+x. A ™. dy = 



21. y = Vl+x 2 A™- dy 



22. y = wax 2 + bx + c. Ans. dy = 



23. y = (a — x)ya+x. Ans. dy = — 



2\/# + # 

xdx 

Vl + x 2 ' 

(2ax + £)dz 
2V«^ 2 +bx + c 

{a + 32)dz 



2V« + 



SB 



x > ' , -, (a — Sx)dx 

24* y - (a + x) ya — * ^4ws, dy = J ; — - — 

2ya — x 



DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 23 

25. y = (l - s* + a; 5 )*. Ans. dy=- — ^ -^ -• 



26. y = (a 2 - x 2 )Va + x. 



Ans. dy = [$(a — 5x)va ■+■ x]dx 

a -f x (3a — x)dx 
37. y = • ^*w. rfy = ~ ~T 

V« — x 2(a — a?)f 



28 .^ = v^S. 4iw. tfy = 



Va — x (a — x)Va 2 — x* 

* 9 - 2/= j/* - Va*^P. 



(z + Va 2 — x 2 )dx 
Ans. dy — 



Wa 2 -x 2 (x- Va 2 -* 2 ) 

i (l+x 2 —4:X*)dx 
80. y = x(l + &■) (1 -*»)*. ^s. ^ = ——! ->- 

31. y= (1 + 2z 2 ) (l + 4a»). 

^4ws. dy = 4z(l -f 3# + 10z 3 )da,. 



32. y = Ans. dy = 



+ V 1 - z 2 2x(l-x 2 )+Vl-x 2 

(1 - Sx)dx 



33. y = (1 + »)Vl — * ^^5. dy — ^- 

l y\ — x 

** A 1 3X * dX 

34. y — — =. Ans. dy = — - 



H DIFFERENTIAL CALCULUS. 






35. y = A/TE^i. Ans. dy = 
V l + ^Tx 



y^ 2(1 + -y^) v'ff — a;t 



36- y - |/2z-l - ^2a: - 1 - V^a - 1 - etc., ad inf 

Squaring, y 2 =2x-l- \Zfce- 1 - ^z- 1 -etc., at/ iw/ 
Hence, # 2 = 2a; — 1 — y, or #* + y = 2x — 1. 



Solving, y = - |± V2z -1 + J = - ^ ±^V8z - 3. 



7 . 1 4tfo 2rfo . 

.-. dy = d: - — = =fc , ^ins. 

2 V8;c- 3 VSx - 3 



III. Differentiation of Transcendental Functions. 

Definition of a Transcendental Function. 

14. A transcendental function is one in which the rela- 
tion between the function and its variable cannot be ex- 
pressed by the ordinary operations of algebra. 

Transcendental functions are divided into three classes* 
Logarithmic, in which the relation is expressed by loga- 
rithmic symbols ; Exponential, in which the variable enters 
an exponent; and Circular, in which the relation is ex 
pressed by means of trigonometric symbols. Thus, 

y = log (ax 2 + b), is a logarithmic function, 

y = (ax 2 + cy x ~ c , is an exponential function, 

and y = sin 2 z — 2tanz, is a trigonometric function 



TRANSCENDENTAL FUNCTIONS. 25 

Differential of a Logaritnm. 
15. It is shown in Algebra that 

log (1 + y) = M(y -«j + | 3 - £ + etc.) (1) 

In which M is the modulus of the system and y any quan- 
tity whatever. 

ds 
Substitute for y the quantity — , s being a function of z, 

s 

and we have, after reduction, 
. (s + ds\ (ds ds 2 ds* , \ , ox 

But the logarithm of a quotient, is equal to the logarithm 
of the dividend, diminished by the logarithm of the divi- 
sor ; changing the form of the first member and suppress- 
ing all the terms in the second member, after the first, 
(Art. 6), we have, 

ds 
log (s + ds) — log s = M — (3) 

But the first member is the difference between two con- 
secutive values of log s ; it is therefore the differential of 
the logarithm of s ; hence, 

d(\ogs) = M^ (4) 

That is, the differential of the logarithm of a quantity is j 

equal to the modulus into the differential of the quantity, \ 

divided by the quantity. ' 

In analysis, Napierian logarithms are almost always used 



26 DIFFERENTIAL CALCULUS. 

Denoting the Napierian logarithm by I, and rememberma 
that the modulus of this system is 1, we have, 

*w = * (5) 

The base of the Napierian system is represented by the 
letter e, and its logarithm in that system is equal to 1. 



Differential of an Exponential Function. 

16. Let a be a constant quantity, s any function of «, 
and assume 

2/ = « s (1) 

Taking the logarithms of both members of (1), we have, 

ly = sla (2) 

Differentiating (2), we have, 

^- = laXds (3) 

y 

Replacing y by its value, and reducing, we have, 
d(a s ) = cflads (4) 



Hence, to differentiate a quantity formed by raising a con- 
stant quantity to a power denoted by a variable exponent, , 
multiply the quantity ly the logarithm of the root and the 
differential of the exponent. 
Again, let us have, 

y = t? (i) 

m which both ^ and s are functions of x. Taking the 
logarithm of both members, we have, 

ly = su (2) 



TRANSCENDENTAL FUNCTIONS. 27 

Whence, by differentiating both members, 
d JL=Uds + s d 4 (3) 

y t v 

Replacing y by its value, clearing of fractions and reducing, 
we have, 

d(t 8 ) = fit da + st 8 ~ x dt (4) 

That is, to differentiate a quantity formed by raising a 
variable quantity to a power whose exponent is variable, 
differentiate first as though the root alone were variable, 
then as though the exponent alone were variable, and take 
the sum, of the results. 

EXAMPLES. 

1. Let y = 1 1 ) = I (a + x) — I (a — x) ; 

\a x/ 

, _ d(a + x) d(a — x) dx dx 



a + x a — x a + x a - x 

2adx 



_. Am. 

a 2 — x 2 




/ r=ic =2 \ l{ - l+x) -\ l ^- x] 



, 1 dx 1 dx dx . 

,\ dy = -- t- -- ■ = - t -. Ans 

* 21+2 21 — a 1—x 2 



X fjnj 

8. liet y — (a) e ; .*. ly = e x la, and — = lae r ledx. 

y 



X 

But, le = 1, .*. dy = (a) e ePladx. Am. 



588 DIFFEKENTIAL CALCULtJS. 

4. Let y = l{lx) = Vx. 

By the rule, dy = -V— * = -r - . ^^* 
J ' 9 Ix xlx 

xdx 
b. y = x l(a + x). Ans. dy = l(a + x)dx ^ — — . 

2^# 

6. y = l(a + a;) 2 = 2Z(a + x). Ans. dy — •. 

7. y = e°(x — 1). Ans. dy = e x xdx. 

8. y = e?(x* -2x + 2). Ans. dy = e x x*dx. 



-1 %e x 



9. y = . Ans. dy = dx. 

*+\ (e x +l)* 



10. y = e x lx. Ans. dy = e x llx + \ 

11. y = l(x + a + V2acc + x 2 ). Ans. dy = 

V%ax + x 2 



\dx 

dx 



e x xe x dx 

12. y = . Ans. dy = ,— ■ — rr. 

* 1 + x * (1 + x)* 

% A -, dx dx 

13. y = I . .4ws. d t/ = 



:Z= • J.A-IVO. VC/W , < 

V* 2 + 1 + x x ■ Vz 2 + 1 

H. y = fy r _y Ans. dy = e —^ j/ j— ^ 

15. y = #"• Ans. dy - ^{Ix + l)dx. 

16. y = a Z:r . Ans. dy = a^" lax" l dx, 



TBANSCEN DENTAL FUNCTIONS. 29 

Differentials of Circular Functions. 
17. Assume the equation, 

y = sina; (1) 

Giving to x the increment dx, and developing, we 

have, 

y + dy — sm(x + dx) 

= sina; cosdx + cosz sindx .... (2) 

Because the arc dx is infinitely small, its cosine is equal 
to 1, and its sine is equal to the arc itself. Hence, 

y + dy = sina; ^ cosxdx (3) 

Subtracting (1) from (3), and replacing y by its value, 
we have, 

d(smx) = cosxdx \a) 

Equation (a) is true for all values of x ; we may there 
fore replace x by (90° — x) ; this gives, 

d[sin(90° - x)] = cos(90° - x)d(90° - x) (4) 

Reducing, we have, 

d{cosx) = — sinxdx (b) 

We have, from trigonometry, the relation, 



, sma; , 

tana; = (5) 

cosa; v 



Differentiating both members, we have, 

, /A x cosa*£(sinz) — simxd (coex) 
* ' cos 2 a; 






30 DIFFERENTIAL CALCULUS. 

Performing indicated operations, and reducing by th* 
relation, sin 2 # + cos 2 x = 1, we have, 

d(ta,nx) = — — (c) 

v ' cos 2 # ' 

Replacing x, m formula (c), by (90° — x) and reducing 
we have, 

We have, from trigonometry, 

ver-sinz = 1 — cos# (6) 

co-versinz = 1 — sinz (7) 

Differentiating (6) and (7), we have, 

c?(ver-sina?) = smxdx (e) 

<tf(co-versinz) — — cosxdx (f) 

We have, from trigonometry, 

sear = (8) 

cos# 

cosecz = - — (9) 

smz v ' 

Differentiating (8) and (9). we have, after reduction, 

d(secx) = tana: secxdx (g) 

d(cosecx) = ~ cote cosecxdx (h) 

The lettered formulas, from (a) to (h) inclusive^ are 
sufficient for the differentiation of any direct circulai 
function. 



TRANSCENDENTAL FUNCTIONS. 31 

EXAMPLES. 

1. Let y = sinra#. 

dy = cosmxd(mx) = m cownx . dx. 

2. Let y = sm w a;. 

dy = m sin m ~ xdsmx — m &m m ~ x cosxdx. 

3. Let y = sin2# cosz. 

dy '■= (dsin2x)(cosx) -\- (dcosx)(sm2x) 

= 2cos2:r cos£ dx — sin&c sina; efo. 

4. Let «/ = Z(sin 2 #) = 2lsmx. 

7 , <#sin£ 2cosz 7 _ , _ 
dy — 2 .—. = — : — e£c = 2cot# dx. 

° %mx sma; 

5. Let y = l^-——y = * /(1 + C(m) ~ * ?(1 ~ C0Sa;) • 

_ 1 smxdx 1 sincc^E sina^a; dx 

** ~ 2 l+cossc 2 1 — cosz 1— cos 2 # sina:' 

6. Let y — e x cosx. 

dy = e x d(cosx) + cosxd(e x ) = /(cosa* — sm#) dx. 

7. y — zsina + cos x. Ans. dy = xcosxdx, 

8. y = 2a-sma; + (2 — £ 2 )cosa;. Ans. dv = a: 2 sina:tf?a:. 
9 y = tana; — a*. Ans. dy — tan 8 ^^. 

[0. y = e cos ' T sina: .:Us. fi?# = e coa2; (cosz — sin 2 :s)da;. 

/ cota;\ 

1 1. u = 2Z(sinz) 4- cosecz. ^^.s. ^z/ = ( 2cot.c : — \dx 

' sina;/ 



32 



DIFFERENTIAL CALCULUS. 

Ans. 



12. y = l(coax + V — 1 sinz) 

13 . y = l(\±^ 

14. y = Z[tan(45° + fc)]. 

15. # = sin(fo). 



a/^Ic/z 



a cosz 



^4ws. cfy = ■ 

3 cos.c 



Ans. dy = - cos (fa;) <#r. 



Differentials of Inverse Circular Functions. 

18. It is often convenient to regard an arc as a function 
of one of its trigonometrical lines. Such functions are 
nailed inverse circular functions, and are expressed by 
such symbols as the following : 

sin - y, cos - y, tan - y, etc., 

which are read the arc whose sine is y, the arc whose cosine 
is y, the arc whose tangent is y, etc. 

Formulas for the differentiation of inverse circular func- 
tions may be deduced from the lettered formulas of the lasf 
article. Thus, from formula (a), we find, 



cosz 



(1) 



1 f we make sim = y, we have, 

x = sin y, and cosz = a/T 

Substituting these in (1), we have, 

dy 



*»• 



<2(sin y) = 



Vi - .v 2 



(•') 



Transcendental functions. &3 

In like manner, from formula (#), we find, 

dx = ^ '- . . . . (2) 

sm# x ' 

Making cosa; = y, we have, 

x — cos - y, and sinz = Vl — y 9 ; 
Substituting these in (2), we have, 



floog-V)-- Jg_ (8») 

vl — ?/ 2 

By a similar course of reasoning we find from the re- 
maining lettered formulas of Art. 17, the corresponding 
formulas lor inverse circular functions, as given below: 



—1 ty 
d(tan l y) = x + , (c 1 ) 



-1 



^(cot-^) - - nf^i w 

^( V ersm~V)= ~7=p • (•*) 

rf(coversm-V) = - ;^== If 1 ) 



1 ^/ 

* M0 y = ^7^fi <**> 



^cosec" »y) = - — ^== . . (*« ) 



9* 



3* DIFFERENTIAL CALCULUS. 

EXAMPLES. 

1. Let y = cos -1 a;Vl — x 2 = cos" 1 \/x* — z«. 



, r~„ r (1 - 2x*)dx 

d Vx 2 — x 4 = * '— 

Vl — x 9 



••• dy=- 
2. Let ^ = sin 



and y/i _ (a-e _ x ^ = *JT^~x* -f z« 
(1 —2a; 8 ) da; 



V(l - x 2 ) (1-x 2 + a;*; 
1 a 



Vl + a; 2 



^'(•fcO+tA-fT 



<fo 1 cfa; 



(! + *■)• '(l + a; 2 )* + **> 
3. Let y = sin _1 2a;v / l —a; 2 . 

dy = d(2xVT^x 2 ) ~- Vl - 4r 2 (l - a* 8 ) 

2(1 - 2a; 2 )tfa; „ , ox 2<fa 

= — — h (1 — 2a- 2 ) 



Vl - x 2 Vl —a: 

. _i 2a; 2<£e 

4. y — tan -. Ans. ay — 



1 - a; 2 * 1 + a;* 

. _il — x 9 . %dx 

5. V — Sill : ^. AWS. «?/ = — r 

* 1 + a:» 1 -I- a; 8 



DEVELOFHENT OF FUNCTIONS. 31 

6. y = \/a 2 — x 2 + tfsin — . Ans. dy = I I dx. 

a * \a + xf 



„ 1 . __i&£ . , dx 

7. y — — versm — . ^ws. eft/ - 



2 9 V9z — 4z 8 



8. y = sec 2#. ^4^s. e?t/ = 



zV^z: 2 — 1 



IV. Successive Diffekentiation and Development 
of Functions. 

Successive Differentials. 

19. The differential obtained immediately from the func- 
tion is the first differential of the function ; the differential 
of the first differential is the second differential of the func- 
tion ; the differential of the second differential is the third 
differential of the function, and so on ; differentials thus 
obtained are called successive differentials, and the opera- 
tion of obtaining them is called successive differentiation. 
If a function be denoted by y, its successive differentials 
will be denoted by the symbols dy, d 9 y, d 3 y, etc. Thus, 
if y = ax 3 , we have, by successive differentiation, dx be- 
ing constant, dy = 3ax 2 dx, d 2 y = Gaxdx 2 , d 3 y = 6adx*. 
d*y = 0. 

Successive Differential Coefficients 

20. If the differential of a function be divided by the 
differential of the variable, the quotient is the first differ- 
ential coefficient of the function ; the differential coefficient 
of the first differential coefficient is the second differential 
coefficient of the function, and so on. Differential coeffi- 



56 DIFFERENTIAL CALCULUS. 

cients, deriyed in this manner, are called successive differ- 
ential coefficients. If a function of x be denoted by y, 
its successive differential coefficients are denoted by the 

symbols, -—, -^ , -=-%. etc. Thus, if we have, as before, 
J dx dx 2 dx z 

y = ax 3 , we have, from the principles just explained, 

It is to be observed that the successive differential 
coefficients are entirely independent of the differential of 
the variable ; so long therefore as this is infinitesimal, the 
differential coefficients will in no way be affected by any 
change in its absolute value. 



EXAMPLES. 

H'ind the successive differential coefficients of the follow- 
ing functions : 

1. y = ax n . 

Ans. ^ = nax n ~ 1 ; p£ = n (n -l)ax n -*; 
dx ' dx 2 v ' 

^ = n(n - l)(ft - 2)ax n ~* ; etc 

if n is a positive whole number, there will be a finite 
number of successive differential coefficients; otherwise 
their number is infinite. 

2. y = ax* + bx 2 . 

Ans. -¥ = 3ax 2 + 2bx; -r r ¥=6ax + 2b; 
dx dx 2 



3=*>2- ft 



DEVELOPMENT OE ETOCTIONS. 3? 

3. y = a x . 

— = a *(la) n . 

dx n 

4. y = since. 

dy d 2 y 

Ans. -r = cosz: -^-f- = — sm x ; 

<# 3 y d*y 

— £. = — cosz ; -=-2- — siiii ; etc. 

ax* ax* 

5. y = Z(aj + 1). 

g = «(. + l)-« ! g--e(. + !)-*;-» 

6. y = sctf*. 



^-* 2 = (• + iK; S = (*+*)' 






McLaurin's Formula. 



21. McLaurin's formula, is a formula for developing a 
function of one variable into a series arranged according 
to the ascending powers of that variable, the coefficients 
being constant. Let y be any function of x, and assume 
the development, 

y = A + Bx+ Cx* 4- Dx 3 + Ex* + etc (1) 



38 DIFFERENTIAL CALCULUS. 

It is required to find such values for A, B, 0, etc., as will 
make the assumed development true for all values of x 
Differentiating, and finding the successive differential co 
efficients of y, we have, 

^ = B + %Qx + Wx* + ±Ex* + etc. ... (2) 

-J[ = 1.2G + 1.2.SDx + 3A£x< + etc. ... (3) 

^ = 1.2.3/? + 1.2.3.40k + etc (4) 

etc., etc., etc. 

But, by hypothesis,, the value of y, and consequently the 
values of its successive differential coefficients, are to be 
true for all values of x ; hence, they must be so for x = 0. 
Making x = 0, in equations (1), (2), (3), etc., and denoting 
what y becomes under that hypothesis by (y) ; what 

2 becomes by (J) ; what g becomes by (g), and so 

on; we have, 

(y) = A ; .-. A = (y) ; 

©-*> ■■■*=(%)> 



(a— *»-iffl). 



etc., etc., etc. 



DEVELOPMENT OE FUNCTIONS. 39 

Substituting these values in (1), we have, 

»-<»> + (S-;+(S)S* (3) ■*.+••*■« 

which is the formula required. Hence, to develop a 
function of one variable in terms of that variable, find its 
successive differential coefficients ; then make the variable 
equal to in the function and its successive differential 

coefficients, and substitute the results for (y), ( y- )» ( -y^ )> 

etc., in formula (5). 

Thus, let it be required to develop sinz into a series 
arranged with reference to x. We have, 

y = sinz, .-. (y) = 

at = cos *< ••• (S) = x 

2 = - sin * •■• (2) = ° 

g = - cos,:, ,. (g) = - i 

etc., etc., etc. 

Hence, 

Sin * = X ~ U3 + L^l5 ~ 1.2.3.45.6.7 + et °" ' " ^ 
In like manner we find, 

cos.? = 1 - ~ + — ^— - ^ + etc (7) 

1.2 1.2.3.4 1.2.0.4.0.6 

It is to be observed that (7) may be found from (6) by 
differentiating and then dividing by dx. 



40 DIFFERENTIAL CALCULUS. 

McLaurin's Formula enables us to develop any function 
of one variable when it can be developed in accordance 
with the assumed law. But there are functions which 
cannot be developed according to the ascending powers of 
the variable ; in this case the function, or some of its sue* 
cessive differential coefficients, become oo, when the variable 
is made equal to 0. 

As a general rule, when the application of a formula 
gives an infinite result, the formula is inapplicable in that 
particular case. 

EXAMPLES. 

I. y = (a + x) n . 

A n n n-\ , n(n — 1) n -2 2 

Ans. y = a + — a x H — — — '-a x* 

1 \.Z 

%y = l(l + x). 

X 2 . X 3 X* x b 

Ans. 2/ = a; "2~ + 3--4- + 5 etc - 

3. y = a x . 

n , (la) (la) 2 9 (la) 3 , (lay 

4. y = Vl + x* = (1 4- x 2 )?. 

Make x % — z, and develrp; then replace z by its value, 

„ x 2 x 4 x* 5x s 



DEVELOPMENT OF FUNCTIONS. 41 



5. y = e 8mx . 



1.2.3.4 1.2.3.4.5 
Sx* 



1.2.3.4.5.6 
6. y -s=xe*+e x -l. 



+ etc. 



3z 2 4a; 3 5a; 4 

Ans.y = 2x + -+—+—J + etc 



Taylor's Formula. 

22. Taylor's Formula is a formula for developing a 
function of the sum of two variables into a series arranged 
according to the ascending powers of one, with coefficients 
that are functions of the other. 

LEMMA. 

If u' = f(x -V y), the differential coefficient of u' will be 
the same, whether we suppose x to remain constant and y 
to vary, or y to remain constant and x to vary, for the 
form of the function is the same, whichever we suppose tc 
vary ; and it has been shown, in Art. 20, that the value of 
the differential coefficient is independent of the value of 
the differential of the variable. Hence, if x + y be in- 
creased either by dx or by dy, and the differential coeffi- 
cient taken, the result will be the same, which was to be 
shown. 

Let u' be a function of (x + y), and assume the devel- 
opment, 

u' = P+Qx + Rx* + Sx* + Tx* + etc (1) 

in which P, Q, R, etc., are functions of y. It is required 
to find such values of P, Q, R, etc., as will make equation 



4:2 DIFFERENTIAL CALCULUS. 

(1) true for all values of both x and y. Since the assumed 
development is to be true for all values of x and y, it must 
be so for x = 0. Making x = in (1), and denoting what 
ik, becomes under this hypothesis, by u, we find, 

P = u. 

That is, P is what the original function becomes, when 
the leading variable is made equal to 0. 

Finding the differential coefficient of u', under the sup- 
position that x is constant and y variable, we have, 

du' dP dQ dR . dS . 

-=- = -=- + -j^x +^-x z + -r-x* + etc (2) 

dy dy dy dy dy v 

Again, finding the differential coefficient of u\ on the 
supposition that y is constant and x variable, we have, 

^- = Q + 2Ex + 3>Sx* + 47V + etc (3) 

cix 

But, the first members of (2) and (3) are equal, by the 
lemma; hence their second members are also equal. If 
we place them equal we have an identical equation, because 
it is true for all values of x, and consequently the coeffi- 
cients of the like powers of x in the two members are 
equal to each other. Placing their coefficients equal, we 
have the following results : 



V dy 


*-dy> 


dy 


1 dHt, 
•'" ~ 1.2 dy* 


3S = d * 

dy 


1 d*u 

■"' ~" L2.3 dy* 


etc.. 


etc.. etc. 



DEVELOPMENT OP FUKCTIOKS. 43 

Substituting the values of P, Q, R, S, etc., in (1), we 
have, 

du x d 2 u x 2 d 3 u x 3 



dy 1 dy 2 1.2 dy* 1.2.3 

d*u x* 

+ T7 • 71TT7 + etc ( 4 ) 

eft/ 4 1.2.3.4 v ' 

which is the required formula. Hence, to develop any 
function of the sum of two variables, we make the leading 
variable equal to 0, and find the successive differential 
coefficients of the result; then substitute them in the 
formula. 

Thus, let it be required to develop (as 4- y) n into a 
series arranged according to the ascending powers of y 
Making y = 0, we have, 

u =x n , 



du n * 

dx 



= nx , 






_ = »(n-l) (rc-2)s n d , 



etc., etc., etc. 

Substituting these in (4), we have, 

/ \n t) n 77—1 n.n — 1 Mi _o _ 

(a; + y) n = x n + ^x n ~ Y y 4- 12 x y 

n(n — 1 hi— 2) ,,_q _ 
+ -* yjs x y + etc " 



44 DIFFERENTIAL CALCULUS. 

This is the binomial formula, in which n is an} 
constant. 

The formula is applicable to every function of the sum 
of two variables; but it sometimes happens that certain 
values of the variable entering the coefficients make the 
first term or some of its successive differential coefficients 
infinite ; for these particular values, the function cannot 
be expressed by a series of the proposed form. 

EXAMPLES. 

Develop the following functions in terms of y. 

1. u' = sm(x + y). 

Making y — 0, we have, u = sinz ; 

du cPu 

.\ -r- = cosz ; -7-^ = — smz ; etc. ; 
dx dx 2 

V • V 2 V s 

nence, u = since + cos?; j — since ^— — cosre-^r-^ 

+ sina; LSr4- etc 

Making x = 0, whence since = 0, and coscc = 4- 1, we have, 
n " = smy = y - ^ + YrsT5 + etC '' aS alread y shown - 

2. u = l(x +y). 

X u = a x+ y. 

Am. u' = a x {\ + (la)y+ { ^y* +ff^ 3 + etc.). 



IMPLICIT FUNCTIONS. 

4. u' = cos(z + y). 



45 



Ans, u' = cos# -- sina; 



y 



co&x 



1.2 



etc. 



v. differentiation of functions of two variables, 
and of Implicit Functions. 

Geometrical Representation of a Function of two Variables. 

23. It may be shown, as in Art. 3, that every function 
of two variables represents the ordinate of a surface of 
which those variables are the corresponding abscissas. If 
the vertical ordinate be taken as the function, it may be 
expressed by the equation, 

■ =/fry) (i) 

In this equation, x and y are independent variables, and 
each may vary precisely as though the other were constant. 
To illustrate, let PQR 
be the surface whose 
equation is (1). 

If, for a given value 
of y as OB, we suppose 
x to vary, equation (1) 
will represent a section 
of the surface parallel 
to the plane xz and at 
the distance OB from 
it. If for a given value 
of x, as OA, we sup- Fig#2 . 

pose y to vary, equa- 
tion (1) will represent a section parallel to the plane yt 




46 DIFFERENTIAL CALCULUS. 

and at the distance OA from it. These sections will be 
called parallel sections, and the corresponding planes will 
be called planes of parallel section. The planes A CD 
and BEF determine the ordinate KK' = z. Let A CD 
A' CD', be consecutive planes of parallel section, at a dis- 
tance from each other equal to dx, and BEF, B'E'F', also 
consecutive planes of parallel section, at a distance from 
each other equal to dy. These planes determine four 
ordinates KK, LL', NN 7 , and MM', which may be de- 
noted by the symbols z, z', z", and z"\ Of these, z and z' 
are consecutive when x alone varies, z and z" are consecu- 
tive when y alone varies, and z and z'" are consecutive when 
both x and y vary. 

Differentials of a Function of two Variables. 

24. The difference between two consecutive states of the 
function when x alone varies is called the partial differ- 
ential with respect to x, and is denoted by the symbol 
(dz) x ; the difference between two consecutive states when 

y alone varies is called the partial differential with respect 
to y, and is denoted by the symbol (dz) \ the difference 

between two consecutive states when both x and y vary ig 
called the total differential, and is denoted by the ordinary 
symbol, dz without a subscript letter. Let us assume the 
figure and notation of Art. 23. We have, from the defini- 
tion, 

z' = z + {dz) x ; whence, by differentiation, 

Wy=(^+(^)„, («) 

The symbol {d 2 z) x indicates the result obtained by dif- 
ferentiating z as though x were the only variable, and then 



TMPLICIT FUNCTIONS. 47 

differentiating that result as though y were the only varia- 
ble, the order of the subscript letters indicating the ordei 
of differentiation. From what precedes, we also have, 

*"'=*'+ {dz') y ; 

substituting for z' and (dz) , their values taken from (2), 
we have, 

«'" = « + (*)„ + (&), + (*•*)„; 

transposing z to the first member, replacing z'" — z by its 
value dz, and neglecting the part (d 2 z) x „, because it is an 

infinitesimal of the second order, we have, finally, 
dz = {dz) x +(dz) y (3) 

Hence, the total differential is equal to the sum of the par- 
tial differentials. 

KXAMPLES. 

1. z = ax 2 + by 3 . 

(dz) x — 2axdx; (dz) = '6by 2 dy .-. dz — 2axdx + dby 9 dy. 

2. z = ax 2 y$. Ans. dz = 2ay 3 xdx + 3ax 2 y 2 dy. 

3. z = $. Ans. dz — yx y dx 4- ofllxdy. 

Notation employed to designate Partial Differential 
Coefficients. 

25. From the nature of the case, there can be no such 
thing as a differential coefficient of a function of two va- 
riables; but the quotient of a par+ial differential oy the 



48 DIFFERENTIAL CALCULUS. 

differential of the corresponding variable, is called a par- 
tial differential coefficient. The form of the symbol indi- 
cates the variable with reference to which the function has 
been differentiated, and no subscript letter is required 
Thus, in Example 2, Art. 24, we have, 

— = 2ay 3 x, and y- = 3ax 2 y 2 . 

A similar notation is employed for partial differential 
coefficients of a higher order, as will be seen in the follow- 
ing article. 

Successive Differentiation of Functions of two Variables. 

26. The second differential of a function of two varia- 
bles is found by differentiating the first differential by the 
rules already given. The third differential comes from the 
second in the same manner that the second comes from 
the first, and so on. In finding the higher partial differen 
cials, the result obtained by differentiating the function 
first with respect to x, and that result with respect to y, is 
the same as though we had differentiated the function first 
with respect to y, and the result with respect to x. For 
from Art. 24, we have, 

z'" = z' + (dz) y = z + {dz) x + (dz) p + (d*z) xy . 

And in like manner, we have, 

*'" = z" + (dz") x = z + (dz) y + (dz) x + {dH) Si m 

Equating these values of z"\ and reducing, we find, 



IMPLICIT FUNCTIONS. 4$ 

which was to be shown. From equation (4), by an exten- 
sion of the notation in Art. 25, we haye, 

d*z _ d*z or d \dx) _ d \ty) 
-dxdy ~ dydx dy dx * ' 

From what precedes, we have, 

dz= (dz) x + (dz) y \ 



EXAMPLES. 

I. Given, z — x 3 y 2 , to find d 2 z. 

( d * z )x, x = ?>y*xdx 2 ; {dH) xy = 6x 2 ydxdy; 

.: d 2 z = 6y 2 xdx 2 + \%x 2 ydxdy + tx*dy*. 
I. Given, z = y 3 x?, to find d z z. 

Am. d 2 z = ~y 3 x^dx 2 + z(-£ij 2 xi)dydx + 6x§ydy* 



ftO DIFFERENTIAL CALCULUS. 

Extension to three or more Variables 

27. If we have a function of three or more independent 
variables, we may find its differential by differentiating 
separately with respect to each variable, and taking the 
sum of the partial differentials thus obtained. The second 
and higher differentials are found in an entirely analogous 
manner. 

Definition of Explicit and Implicit Functions. 

28. An explicit function, is one in which the value of 
the function is directly expressed in terms of the variable. 
Thus, y —\/2rx — x 2 , is an explicit function. An implicit 
.unction, is one in which the value of the function it 
not directly given in terms of the variable. Thus, in the 
equation, 

ay z + bxy + ex 2 + d = 0; 

y is an implicit function of x. Implicit functions are 
generally connected with their variables by one or more 
equations. When these equations ar*e solved the implicit 
function becomes explicit. 

Differentials of Implicit Functions. 

29. The differential of an implicit function may be 
found without first finding the function itself. For, if 
we differentiate both members of the first equation, we 
shall thus find a new equation, which, with the given one, 
will enable us to find either the differential, or the differ- 
ential coefficient of the function. For example, suppose 
we have the equation, 

y * +- 2xy + x 9 - a* - . . . . (1) 



xMPLICIT FUNCTIONS. 51 

to find the differential coefficient of y. Differentiating 
(1), and dividing by 2, we have, 

ydy + xdy + ydx + xdx = (2) 

du 
Finding the value of -j-, we have, 

^ = -1. 

dx 

Again, let us have the relation, 

xy = m (1) 

du 
to find the value of -~. Differentiating (1), we have* 

xdy + ydx = (2) 

Whence, 

dx x 



(3) 



But from (1), we have, 



y__m 
x x 2 ' 



Substituting in (3), we find, 

dy _ in 
dx x s 



(4) 



EXAMPLES. 



Find the first auiu. second differential coefficients of y 
in the following" imolicit functions : 



52 DIFFERENTIAL CALCULUS. 

1. y* — By + x = 0. 
We have, 

3,^-3^ + ^ = 0, ,.|= l.^l-j 

also, 

6^ 2 + 3# 2 d 2 ?/ - 3d 2 y= 0, 

(Py _ _2y_ dy 2 __2 y 

""" dz 2 ~~ l-?/ 2 " S« ~~ 9 ' (1-y 8 ) 1 ' 

2. # 2 - touqy + x 2 — a = 0. 

Ans. ^ = ^y-^ and ^ = g(w>i ~ 1} 
* dx y — rnx f dx 2 ~ (y — ma;) 1 



PART II. 

APPLICATIONS OF THE DIFFEKENTIAL 
CALCULUS. 



K/ 




/T 
L 



I. Tangents and Asymptotes. 

Geometrical Representation of the First Differential 
Coefficient. 

30. It has been shown (Art. 3), that any function of 
one variable maybe represented by the ordinate of a curve, 
of which that variable is the cor- 
responding abscissa. We have also 
seen (Art. 5), that an element of 
the curve, PQ, does not differ from 
a straight line. This line, prolonged 
toward T, is tangent to the curve 
at P. The angle that the tangent 
makes with the axis of X is equal 
to RPQ ; denoting it by 6, and 
remembering that the tangent of the angle at the base 
of a right angled triangle is equal to the perpendicular 
divided by the base, we have, 



AB 



Fig. 3. 



tan 



dx 



The tangent of is taken as the measure of the slope, 
not only of the tangent, but also of the curve at the point 
P. Hence, the slope of a curve, at any point, is measured 



54- DIFFERENTIAL CALCULUS. 

by the first differential coefficient of the ordinate at that 
'point. 

The slope is positive or negative, that is, the curve slopes 
upward or downward, according as the first differentia] 
coefficient is plus, or minus. 



Applications. 

31. The principle just demonstrated enables us to find 
the point of a curve, at which the tangent makes a given 
angle with the axis of x. 

Thus, let it be required to find the point on a given 
parabola at which the tangent makes an angle of 45° with 
the axis. Assume the equation of the parabola, 

^-■ Zm - . d JL-l 
y ~ lpX ' - dx-y 

Placing this equal to 1, we have, 

P i 

- = 1, or y = p. 

V . 

But y = p, is the ordinate through the focus. Hence, the 
required point is at the upper extremity of the ordinate 
through the focus. 

Again, let it be required to find the point at which a tan- 
gent to -the ellipse is parallel to the axis o£x. Assume the 
equation, 

y 2 x% _ ty _ b*x 

b* + a* ~ ' *'* dx ~ a*y 



Placing this equal to 0, we have, 

a y 



TANGENTS AND ASYMPTOTES. 55 

which can only be satisfied by making x = 0; this, in the 
equation of the curve, gives y = =fc b. Hence, the tangent 
at either vertex of the conjugate axis fulfills the given 
condition. 

Again, to find where the tangent to an hyperbola is per- 
pendicular to the axis; assume the equation of the curve, 



(1) 



a 2 b 2 
wnence, by the rule, 

dy b 2 x . , 

Hence, the tangent at either extremity of the transverse 
axis, fulfills the given condition. 



Equations of the Tangent and Normal. 

32. Let P be a point of the curve whose co-ordinates are 
x", y" ; then will the equation of a straight line through 
it, be of the form, 

V-y" = a(x - x") . . . . (1) 

in which a is the slope. If we make a equal the tangent 

of PPQ, the line will be tangent to the curve. But the 

dy" 
tangent of RPQ is -j-jr\ hence, we have, for the equation 

of a tangent line, 

y-y" =%{*-*") (2) 

If we make a equal to minus the reciprocal of the tan- 
gent of RPQ, the line will be perpendicular to the tangent, 



56 DIFFERENTIAL CALCULUS. 

that is, it will be normal to the curve; hence, we have, foi 
the equation of a normal line, 

y-y"=-§f,(*-*") (3) 

In these equations x and y are general co-ordinates of 
the lines, and x", y" are the co-ordinates of the point of 
contact. 

Let it be required to find the equation of a tangent to 

an ellipse. We found, in the last article, the value of 

dij h 2 or 

-f = r- ; substituting for x and y the particular values 

ctx cl y 

x" and y", and putting the result in (2), we find, 

h 2 x" 

whence, by reduction, 

yy" xx" , ... 

ZJ± — I = l (4) 

Making a = b, in this equation, we have, 

yy" + xx" — a 2 , 

which is the equation of a tangent to a circle. 

To find the equation of a normal to an ellipse we substi 

dx" 
tute the value of -y-77, in equation (3), which gives, 

y-y" = %^-^ < 5 > 

Mak ng a = b, we have, 

y-y"=^,(p=-x"), 

which is the equation of a normal to a circle. 



TANGENTS AND ASYMPTOTES. 57 

If, in equation (2), we make y = 0, the corresponding 
value of x — x" will express the length of the subtangent, 
counted from the foot of the ordinate through the point 
of contact ; denoting this by S.T., we have, 



If, in equation (3), we make y = 0, the corresponding 
value of x — x" will express the length of the subnormal, 
counted from the foot of the ordinate through the point 
of contact ; denoting this by S.N., we have, 

S.N=y" d £, (7) 

dy" 
Substituting in (6) and (7) the values of -^77 taken from 

CLX 



the equation of the ellipse, we have, 



am a2 y" 2 ;j our Wx" 



Asymptotes. 

33. An asymptote to a curve, is a line that continually 
approaches the curve and becomes tangent to it at an 
infinite distance. The characteristics of an asymptote 
are, that it is tangent to the curve at a point that is infi- 
nitely distant, and that it cuts one or both axes at a finite 
distance from the origin. 

To ascertain whether a curve has an asymptote, assume 
the equation of the tangent (Art. 32), and in it make 
z, and y, successively equal to 0. 

3* 



58 DIFPEBENTIAL CALCULUS. 

Id this manner we find, 



4D = y " _ X " ^-„ and 



AT=x" -y 



,,dx" 

w 



a) 



dy" 




We then find -—-, from the 

CtX 

equation of the curve, substi- 
tute it in (1) and make the hypothesis that places the 
point {x", y'') at an infinite distance. If this supposition 
make either AD or AT finite, the curve has an asymptote, 
otherwise not. 

Let it be required to find whether the hyperbola has an 

dy" b 2 x" 
asymptote. We have found (Art.- 31), -j- fl = ^j„, which 

in (1) gives, 
AD = y' 



a 2 y 



t 2 *" 2 z Am ,, «y 2 



«v 



b 2 x' 



(2) 



Whence, by reduction, 



. j, a 2 y" - b 2 x" . . m b 2 x" - a 
U> = — ,777 , and AT = -^ 



2 „ 8 



a 2 y 



(3) 



b 2 a 2 

or, AD = r, ? and A T =—r, 



W 



The only hypothesis that places a point of the hyperbola 
At an infinite distance is y" = d= oo , and x" = =fc go ; both 
these sets of values, ir. (4), make AD and AT equal to 0. 
Hence, the hyperbola has two asymptotes, both passing 
through the centre. 



TANGENTS AND ASYMt'TOTES. 5S 

To find whether the parabola has an asymptote, make 

~r~n = v m (I)? whence, 
ax y 

AD = y>-P^ =1^1, and AT = -x". 

y" y 

If we make y" — oo, and x" = oo, to put the point oi 
contact at an infinite distance, we find both AD and A T 
equal to oo , which shows that the parabola has no asymp- 
tote. 

Order of Contact. 

34. Two lines have a contact of the first order, when 
they have two consecuth T e points in common. The first of 
these is the point of contact ; this point is common to both 
lines, and, as we have jnst seen, the first differential coef- 
ficients of the ordinates of the two lines at this point are 
equal. Conversely, if two lines have a common point, and 
if the first differential coefficients of the ordinates of the 
lines at that point are equal, the lines have a contact of the 
first order. 

Two lines have a contact of the second order, when thev 
have three consecutive points in common. The first of 
the three is the point of contact, and because the lines have 
three consecutives ordinates common, the first and second 
differential coefficients of their ordinates at this point art 
equal. Conversely, if two lines have a common point, and 
the first and second differential coefficients of their ordi- 
nates at that point equal, the lines have a contact of the 
second order. 

In like manner it may be shown, if two lines have a 
point in common, and n successive differential coefficients 
of their ordinates at that point equal, that the lines have 



60 DIFFERENTIAL CALCULUS. 

a contact of the n th order, n being any positive whole 
number. 

If a line oe applied so as to cut a given line in an even 
number of points, it will, just before the first, and just 
after the last, lie on the same side of the given line ; and 
this is true when the points of secancy are consecutive, in 
which case the lines have a contact of an odd order. 
Hence, if two lines have a contact of an odd order, they do 
not intersect at the point of contact. If the applied line cut 
the given line in an odd number of points, it will, just 
before the first, and just after the last, lie on opposite sides 
of the given line ; and this is also true when the points of 
secancy are consecutive ; hence, if two lines have a contact 
of an even order, they intersect at the point of contact. 

If the applied line be straight, and the contact of the 
n th order, the given curve will have n consecutive values 
of dy, equal to each other. If each of these be taken from 
the next in order, the differences will be 0, that is, the 
curve will have n — 1 consecutive values of d 2 y equal to 
0. In like manner, we may show that it has n — 2 consec- 
utive values of d'y equal to 0, and so on, to the (n + l) th 
differential of y, which will not be 0, but will be plus or 
minus, according as the (n + l) tb value of dy, counting 
from the point of contact, is greater or less than the n th 
value of dy. Conversely, if the (?i + l) th differential of y 
is plus, the (?i -f l) th value of dy is greater than the n ih ; 
if minus the (n + l) tb value of dy is less than the n th . 

II. Curvature. 

Direction of Curvature. 

35. Let EL be a curve, whose concavity is turned 
downward, that is, in the direction of negative ordi- 



CURVATURE. 



61 




A B 



Fig. 5. 



aates ; let AP, BQ, and GQ' be consecutive ordinates, AB 
and BG being equal to dx; prolong PQ toward T. Then 
will PQ, and R Q', be consecutive values of dy, and con- 
sequently their difference, R'Q' — RQ, will be the value of 
d 2 y at P. The right angled triangles, PRQ, and QB'S, 
have their bases, and the angles at their bases, equal ; hence, 
their altitudes, RQ, 
and R'S, are equal ; 
but R'Q' is less than 
R'S, because Q' is 
nearer — oo than S; 
hence, R'Q' is less 
than RQ, and con- 
sequently the value 
of d 2 y is negative. 
Conversely, if d 2 y 
is negative, R'Q' is 
less than R'S, and the curve in passing from P is concave 
downward. In like manner, it may be shown that the 
curve is concave iqnuard at P, when d 2 y is positive. Be 
cause the sign of the second differential coefficient is the 
same as that of the second differential, we have the follow- 
ing rule for determining the direction of curvature, in 
passing from any point toward the right . 

Find the second differential coefficient of the ordinate at 
the point ; if this is negative, the curve is concave down- 
ward, if positive, it is concave upivard. 

We may regard y as the independent variable, and find 
the second differential coefficient of x ; if this be negative 
at any point, the concavity is turned toward the left, if 
positive, toward the right. 

Example. — Let it be required to determine the direction 
of curvature at any point of a parabola. 



62 DIFFERENTIAL CALCULUS. 

Assume the equation, y 2 = 2px; whence, by differentia 
fcioQ and reduction, 

d*y _ _p 2 _ <Px _ 1 

dx 2 ~ y 3 ' dy 2 ~~ p 

The first of these is negative for positive values of y, and 
positive for negative values of y ; hence, the part of the 
curve above the axis of X is concave downward, and the 
part below is concave upward. The second is positive for 
all values of x and y ; hence, the curve is everywhere con- 
cave toward the right. 

If the tangent at P have a contact of the ?i th order, n 
jeing greater than 1, tne second differential coefficient of 
y at that point will be (Art. 34). In this case it may 
oe shown, as above, that the curve bends downward in 
passing toward the right, when the (n + l) tb value of dy, 
counting from P, is less than the n th value of dy, and up- 
ward when the (n + l) th value of dy is greater than the n th . 
The former condition is satisfied when the (n + l) th differ- 
ential coefficient of y is negative, and the latter when it is 
positive (Art. 34). 

Amount of Curvature. 

36. The change of direction of a curve in passing from 
an element to the one next in order, is the angle between 
the second element and the prolongation of the first. This 
angle is called the angle of contingence, it is negative when 
the curve bends downward, and positive when it bends 
upward. The total change of direction in passing over any 
arc is equal to the algebraic sum of the angles of contin- 
gence at every point of that arc. 

The curvature of a curve is the rate at which it changes 
direction. When the curvature is uniform, as in the 



CtTKVATCTRE. 



6S 



circle, it is measured by the total change of direction in 
any arc. divided by the length of that arc. In any curve 
whatever the curvature may be regarded as uniform foi 
an infinitely small arc; hence, the curvature at any point 
of a curve, is equal to the angle of contingence at that 
point, divided by the corresponding element of the curve. 
Denoting the angle of contingence by d&, the correspond- 
ing element by ds, and the curvature by c, we have, 



ds 



a) 



Osculatory Circle. 

37. An oscillatory circle, to a given curve, is a circle tnat 
has three consecutive points in common with that curve. 
Thus, if the circle whose centre is C, pass through the 
three consecutive points P, Q, and E, of the curve, KQ, it 
is osculatory to that curve at P. 

The circle and curve have two consecutive tangents, PT 
and QT\ and 
two consecutive 
normals, PC and 
QC, in common; 
they have also 
the angle of con- 
tingence at Q, in 
common ; hence, 
they have the 
same curvature 
at the point P. 
Because P C and 
QC are perpen- Fi s- 6 - 

dicular to PT, anl QT, their included angle, is equal tc 




64 DIFFERENTIAL CALCULUS. 

the angle of con tin gen ce, cU; hence, PQ = CP X d\ oi 
denoting PQ by ds and CP by R, we have, ds = Rdd, or, 

*?=! ....(1) 
ds R w 

This equation shows that the curvature of a curve at 
any point is equal to the reciprocal of the radius of the 
oscillatory circle at that point; for this reason, the radius 
of the osculatory circle is called the radius of curvature. 



Radius of Curvature. 

38. If we denote the inclination of the tangent, P7\ to 
the axis of X by $, the angle of contingence, TQT, equal to 

PCQ, will be equal toc$; but we know that & = tan - -f-J 
differentiating, we have, 



dx dxd 2 y 

~ ^ §1 ~ % 2 + dx 2 ' 
+ Tx* 

Finding the value of R from equation (1), Art. 37, ana 
replacing ds by its value, from Art. 5, we have, 

n ~ dxd*y K } 

Dividing both terms of the second member by dx 3 , and 
denoting the first differential coefficient of y by q', and its 
second differential coefficient by q", we have, 

i=£±£2* (2) 



CURVATURE. 65 

When the curve bends downward, d& is negative, and 
consequently R is negative ; when the curve bends upward, 
both are positive. 

It is sometimes convenient to regard R, or some other 
quantity on which R depends, as the independent variable. 
In this case both dy and dx are variable, and we have, 

j a — dxd % y — dyd 2 x t 
= dx* + dy* ; 

and this in (1), Art. 37, gives, 

dxd*y-dyd*x K } 

Example.— Required the value of R for the parabola. 
Assume the equation, y 2 = 2px. 
By differentiation and reduction, we have, 

dy p , d 2 y ., p 2 

-f == q f = S and -=-§ = q" = - Z- ; 

dx y dx 2 y 3 



substituting these in (2), we have, 




«-('+$)'*-$- 


JO 2 



The value of R is least possible when y — 0, increases 
as # increases, and becomes oo when v — oo . Hence, the 
curvature of the parabola is greatest at the vertex, and 
diminishes as the curve recedes from the vertex. 

Co-ordinates of the Centre of the Osculatory Circle. 

39. In the right angled triangle, PHC, PH is perpen- 
dicular to ED, and PC to EP ; hence, the angle, HPC y is 



f)6 DIFFERENTIAL CALCULUS. 

equal to 6. Denoting the abscissa, A C, by a, and the ordi- 
nate, DC, by (3, we have, 

a = AH+ EC, and /3 = BP - HP (1) 

A If and BP are the co-ordinates of P, denoted by x and 
y; HP is equal to PCcosd, and HC to PCsin&; substi- 
tuting for PC, or R, its value found in thcj last article, re- 
membering that it is negative in the case under considera- 
tion, (Art. 38), replacing sind, and cos0, by their values 
taken from Art. 5, and reducing, we have, 



dx 2 + dy 2 dy _ (\ + q' 2 



dx \ q 

dx 2 -f dy 2 1 -}- q'- 



m<i 



(2) 



Locus of the Centre of Curvature. 

40. A line drawn through the centres of the oscillatory 
circles at every point of a curve is called the evolute of that 
curve, and the given curve, with respect to its evolute, is 
called an involute. If we consider any radius of curvature 
of a curve, we see that it intersects the one that precedes 
and the one that follows it, and the distance between the 
points of intersection is an element of the evolute ; hence, 
the radius is normal to the involute, and tangent to the 
evolute. We also see that the difference between two con- 
secutive radii of curvature is equal to the corresponding 
element of the evolute ; hence, the difference between any 
two radii of curvature of a given curve is equal to the cor- 
responding arc of the evolute. 

The evolute of a curve may be constructed by drawing 
normals to the curve, and then drawing a curve tangent 



CURVATURE. 67 

to them all: the closer the normals, the more accurate will 
be the construction. An involute may be constructed by 
wrapping a thread round the evolute, holding it tense^ 
and then unwrapping it. Every point of the thread de- 
scribes a curve, which is an involute of the given evoluie. 
Practically, the evolute is cut out of a board, or other solid 
material. In this manner the outlines of the teeth of 
wheels are sometimes marked out, the circle being taken 
as an evolute. 

Equation of the Evolute of a Curve. 

41. The equation of the evolute of any curve may be 
found by combining the equation of the curve with form- 
ulas (2), Art. 39, and eliminating x and y. The method 
will be best illustrated by an example; thus, let it be re- 
quired to find the equation of the evolute of the common 

parabola. 

p p 2 

We have already found q' = -, and q"— — -— (Art. 38) ; 

substituting these in (2), Art. 39, we have, after reduction, 

a = x + -^ — = 3x + p, >'• x = -(a — p)j 

p 6 



y(y 2 +p 2 ) = _t . . /A„* 

p 2 p' 



= y-^ M -I JLJ = - fe ■•• y = -*V; 



substituting the values of x and y in the equation y 2 = 2px. 
we have, 



,3 2p, ^ . „*_ 8 



which is the equation required. It is the equation oi a 
semi-cubic parabola. 



68 DIFFERENTIAL CALCULUS. 

EXAMPLES. 

1. Find the equation of the evolute of an ellipse 

Ans. (aa)* + (6/3)* = (a 9 - b^ 

2. Find the equation of the evolute of an hyperbola. 

Ans. (aa)* - (Z>/S)* = (a 9 + b 9 )% 

3. Find the radius of curvature of an ellipse. 

« 4 Z> 4 

4. Find the radius of curvature of an equilateral hyper- 
bola referred to its asymptotes, the equation being xy = m. 

(m 9 4- xA)% 

Ans. p = - — ——, 

2mx* 



III. Singular Points of Curves. 

Definition of a Singular Point. 

42. A singular point of a curve, is a point at which the 
curve presents some peculiarity not common to other 
points. The most remarkable of these axe, points of in- 
flexion, cusps, multiple points, and conjugate points. 

Points of Inflexion. 

43. A point of inflexion is a point at which the curva 
ture changes, from being concave downward to being con- 
cave upward, or the reverse. 

Inasmuch as the direction of curvature is determined by 
the sign of the second differential coefficient of the ordi- 



SINGULAR POINTS OP CURVES. 6fc 

nate, it follows that this sign must change from — to +, 
or from + to — , in passing a point of inflexion. But a 
quantity can only change sign by passing through 0, or oo . 
Hence, at a point of inflexion the second differential coef- 
ficient of y must be either 0, or oo . If the corresponding 
values of x and y are such, that for values immediately 
preceding and following them, the second differential coef- 
ficient has contrary signs, then will each set of such values 
correspond to a point of inflexion. 

EXAMPLES. 

1. To find the points of inflexion on the curve whose 

equation is y = b + (x — a) 3 . 

d 2 y 
We find, -j~ = 6(x — a) ; this, put "equal to 0, gives 

x = a, which, in the equation of the curve, gives y = b. 
When x < a, the second differential coefficient is negative, 
and when x > a, it is positive. Hence, the point whose 
co-ordinates are a and b, is a point of inflexion. 

2. To find the points of inflexion on the curve whose 

equation is y = b + T V (x — #) 5 - 

d 2 u 
We find, -^| = 2 O — a) 3 ; this placed equal to 0, gives 

x = a, whence, y — b. When x < a, the second differen- 
tial coefficient is negative, and when x > a, it is positive. 
Hence, the point whose co-ordinates are a and b, is a point 
of inflexion. 

3. Find the co-ordinates of the points of inflexion on 



the curve whose equation is y 



- y E 



Ans. x = -r, and y = ^ r V& 



70 



DttftfEftENTlAL CALCULUS. 



4. Find the co-ordinates of the points Of inflexion on 

x 2 (a 2 x 2 ) 

the line whose equation is y = — - — ^ -. 

1 - 5 

Arts, x = ± -a v'ti, and y = —a. 
6 ^36 

Cusps. 

44. A cusp is a point at which two branches terminate, 
being tangential to each other. There are two species of 
cnsps ; the ceratoid, named from its resemblance to the 
horns of an animal, and the ramphoid, named from its 
resemblance to the beak of a bird. The first is shown at 
E, and the second at A. 

The method of determining the position and nature of a 
cusp point will be best shown by examples. 

1. Let us take the curve whose equation is y=b± (x— a)i 
From this we find, 



dy_ 

dr 



± - (a; -^,and-f = ± J (^ 



a) 



i 



For x = a, -r — 0, and 7 
dx dx 2 



? = oo 



For x < a, both are 



imaginary, as is also the value of y. For x > a, both are 
real, and each has two values, one plus and the other 
minus. Hence, the point whose 
co-ordinates are a and b is a cusp 
of the first species. 

2. Let us take the curve whose 

equation is y — x 2 =b x2. 
From it we find, 



\ 



Pig. 7. 



dx 



2x ± I x%, 



and -y-f = 2± — x*. 
dx 2 4 



SINGULAR POINTS OF CUKVES. 



73 



From the given equation we see that no part of the 
curve lies to the left of the axis of y, and that there are 
two infinite branches to the right of that axis. 

The values of ~ are both at the origin : and at that 
dx G ' 

d 2 y 
point both values of -=-| are positive ; hence, the origin is 

ctx 

a cusp of the second species. 

A discussion of the above equations shows that the 
upper branch has its concavity always upward; the second 

64 



branch has a point of inflexion for, x = 



225 



its slope is 



for x = — ■ ; and it cuts the axis of x 

at the distance 1 from the origin ; from 
the origin to the point of inflexion, 
it curves upward ; after that point it 
curves downward. The shape of the 
curve is indicated in the figure. 

For the ceratoid the values of the 
second differential coefficient of y have 
contrary signs, for the ramphoid they have the same sign. 




Fig. 8. 



Multiple Points. 

45. A multiple point is a point where two or mort 
branches of a curve intersect, or 
touch each other. 

If the branches intersect, -^ will 
dx 

have as many values at that point 

as there are branches ; if they 

are tangent, these values will be 

equal. 




Fig.a. 



?2 DIFFERENTIAL CALCULUS. 

EXAMPLES. 

1. Take the curve whose equation is y 2 = x 2 — xK 

For every value of x there are two values of y equal 
with contrary signs; hence, the curve is symmetrically 
situated with respect to the axis of x. For x — 0, we have 
y = ± ; hence, both branches pass through the origin. 
From the equation, we find, 

dy _x 2x 3 1 2x 2 

dx~y"~Y yT" 17 ^ ^ Vl-x 2 ' 

dv 
For x = 0, — = dz 1, hence the origin is a multiple poim, 

by intersection. The curve is limited in both direc- 
tions. 



2. Take the curve whose equation is y = =b vx 6 + X*. 
The two branches are symmetrically placed with respect 

Co the axis of x, both pass through the origin, forming a 
loop on the left and extending to infinity on the right 
We find, from the equation of the curve, 

dy 5rc 4 + 4# 3 hx 2 + 4z 

dx ~ 2^5 + x * " 2y/ x + i 

For x — 0, we have -^- = dc ; hence, the origin is a mul- 
ax 

tiple point, the branches being tangent to each other and 

to the axis of x. 

/ a % _ x 2 

3. The curve whose equation is y = ±lxA/ g — ^ 

has a multiple point at the origin, and is composed of two 
pointed loops, one on the right and the other on the left 
of the or?s;in. 



SINGULAR POINTS OF CURVES. 73 

4. The curve whose equation is y 2 = (x — 2) (x — 3) 2 is 
symmetrical with respect to the axis of x and consists of a 
looped branch which extends from x = 2 to x = 3 ; at the 
latter point the upper part passes below the axis of x and 
lower part passes above that axis, forming a multiple point ; 
beyond x = 3 the two parts continually diverge, extending 
to x = (/> . 

Conjugate Points. 

46. A conjugate, or isolated point, is a point whose 
co-ordinates satisfy the equation of a curve, but which has 
no consecutive point. Because it has no consecutive point, 
the value of the first differential coefficient of y at it, is 
imaginary. 

EXAMPLES. 



1. Take the curve whose equation is y = =fc x wx — a. 

In this case x = and y — 0, satisfy the equation, but 
no other value of x less than a gives a point of the curve. 
Furthermore, we have, 

dy _ . 1 3x 2 - 2ax 
dx~ 2 Vz 3 — ax 2 ' 

which for x = is imaginary, as also for all values of 
x < a. 

2. Take the curve whose equation is 



y = ax 2 ± a/#(i — cosz). 

For every positive value of x, there are two values of y, 
and consequently two points, except when cosz = 1, when 
the two points reduce to one. These points constitute a 
series of loops like the links of a chain, having for a dia- 



74 



DIFFERENTIAL CALCULUS. 



metral curve a parabola whose equation is y = ax 2 . Foi 
every negative value of x the values of y are imaginary, 
except when cosx = 1; in these cases we have a series of 
isolated points situated on the diametral curve, whosl 
equation is y = ax 2 . We have, by differentiating and 
reducing, 



dy 

dx 



2ax ± 



\/l — cos£ + x^/1 4- cosz 



2Vx 



which is imaginary for negative values of x. 



IV. Maxima and Minima. 



Definitions of Maximum and Minimum. 

47. A function of one variable is at a maximum state 
when it is greater than 
the states that im- 
mediately precede and 
follow it; it is at a 
minimum state, when 
it is less than the states 
that immediately pre- 
cede and follow it. 
Thus, if KL be the 
curve of the function, 
BN will be a maxi- 
mum, because it is 
greater than AM and 
CO, and EQ will be 
a minimum, because it 
is less than DP and 
FR 




ABC DEE 



N 




K 



*S 



Q 



ABC D £ F 

Fig. 10. 



MAXIMA AND MINIMA. 75 

Analytical Characteristics of a Maximum or Minimum. 

48. If we examine the figures giyen in the last article, 
we see that the slope of the curve KL, or the differential 
coefficient of the function, changes from + to — in pass- 
ing a maximum, ami from — to + in passing a minimum. 
But a varying quantity can only change sign by passing 
through 0, or oo ; hence, the first differential coefficient 
of the function must be either 0, or oo , at either maximum 
or minimum state. 

If, therefore, we find the first differential coefficient of the 
function, and set it equal to and oo, we shall have two equa- 
tions which will give all the values of the variable that belong 
to either a maximum or minimum state of the function. 

They may also give other values; hence the necessity 
of testing each root separately. This may be done by the 
following rule : 

Subtract from, and add to, the root to be tested, an infi- 
nitely small quantity ; substitute these successively in the 
first differential coefficient ; if the first result is plus and 
the second minus, the root corresponds to a maximum ; if 
the first is minus and the second plus, it corresponds to a 
minimum; if both have the same sign, it corresponds to 
neither a maximum nor a minimum. 

The maximum or minimum value may be found by sub- 
stituting the corresponding root in the given function. 

EXAMPLES. 

1. Lety = 3 + (x-2) 2 - 

dv 
By the rule, we have, ~ — 2(x — 2) = 0, /. x = 2. 

Scbstituting for x the values, 2 — dx, and 2 + dx, we finO 



76 DIFFERENTIAL CALCULUS. 

for the corresponding values of the differential coefficient 
— 2dx, and + 2dx ; hence, x = 2, corresponds to a mini- 
mum, which is y = 3, 

2. Let y = 4 - (x - 3)1 

We haye, 

| = -§<*-»r*=-; ■"=»■ 

For x = 3 — dx, and 3 + J», we have the differential coef- 
ficient equal to 

2 , 2 

and. — 



hence, x = 3 corresponds to a maximum, y = 4. 

3. Let y = 3 + 2(z-l) 3 - 
We have, 

g = C(*-l)' = 0; ,* = 1. 

Substituting 1 — $z, and 1 + dx, for x, in the first differ- 
ential coefficient, we have in both cases a positive result; 
hence, x = 1 does not correspond to either a maximum, or 
minimum. 

The preceding method is applicable in all cases ; but, 
when the first differential coefficient of the function is 0, 
as it is in most instances, there is an easier process for 
testing the roots. In this case, the function being repre- 
sented by the ordinate of a curve, the maximum and 
minimum states correspond to points at which the tangent 
is horizontal ; furthermore, the curve lies wholly above, or 
wholly beloiv, the tangent at the point of contact, that is, 
the tangent does not cut the curve at that point; hence, 
the contact is of an odd order, and consequently the first, 
of the successive differential coefficients of the function that 



MAXIMA ASTD MIXIMA. 77 

does not reduce to must be of an even jrder (Art. ,34). 
If this is negative, the curve bends downward after passing 
the point of contact, and the root corresponds to a maxi- 
mum, if positive the curve bends upward, and the root cor- 
responds to a minimum (Art. 35). Hence the following 
practical rule for finding the values of the variables that 
correspond to maxima and minima: 

Place the first differential coefficient of tlie function equal 
to 0, and solve the resulting equation; substitute each root 
in the successive differential coefficients of the function, 
until one is found that does not reduce to ; if this is of 
an even order and negative, the root corresponds to a maxi- 
mum, if of an even order and positive, to a minimum ; but 
if of an odd order, it corresponds to neither maximum nor 
minimum. 

EXAMPLES. 

1. Let y = x* - Sx + 2. 
We have, 

^ = 2.-3 = 0; ,.* = §. 
dx 2 



Also, 






Hence, x = -, corresponds to a minimum state, which ig, 

4/ 

y=[x*-Zx + 8] = - y 

rd^v~\ d^ it 

The symbol y^ 3 is used to denote what -j-^ becomes 

when the variable that enters it is made equal to -. 

4 



78 DIFFERENTIAL CALCULUS. 



= _ 4(z-3) 3 = 0; /. * = 3; 



dy 



d 2 y _ _ 10 y /w _Q\ f . . \~d 2 y 

dx' 2 



we also find, 

[g] 3 = o - d [3G 3 =- 34 - 

Hence, [y] 3 = 4, is a maximnm. 

In applying the above rule, a 'positive constant factoi 
may be suppressed at any stage of the process; for it is 
obvious that such suppression can in no way alter the 
value of the roots to be found, or the signs of the successive 
differential coefficients. 

3. Let y = 3z 3 - dx 2 - 27x + 30. 

Rejecting the factor, + 3, and denoting the result by u, we 
have, 

dn 

™ = 3 x z-6x-9 = 3(x*-2x-3) = 0; .-. x = 3,x= - 1. 

Rejecting the factor. + 3, and denoting the resulting value 
of the function by u', we have, 



rd 2 u'-i , rd 2 u'-\ 

[&rJ = 4 > and b^-L = - 4 



The first corresponds to a minimum and the second to a 
maximum. Substituting in the given function, we have, 
for the maximum and minimum values, 

y' — 45, and y" = — 51. 
When the first differential coefficient is composed of 



MAXIMA AND MINIMA. 79 

variable factors, the method of finding the corresponding 
values oi the second differential coefficient may be simpli- 
fied as follows : let us have, 

dx -rxy, 

P and Q being functions of x, and suppose that [P] a = 0. 
By the rule for differentiating, we have, 

dx* ~ dx + V dx> 
but because P becomes when x = a, we have, 

Ute»Ja L v da?Ja 

Hence, ^wrf the differential coefficient of the factor that 
reduces to 0, multiply it by the other factor, and substitute 
the root in the product. The result is the same as that 
found by substituting the root in the second differential 
coefficient, 

4. Let?/ = (x - 3) 2 (x- 2). 

We have, 

^ = 2(x - 3) (x - 2) + (x - 3)2 = (x - 3) (3* - 7) = ; 

7 

.'. X = O, # = — . 

By the principle just demonstrated, 

4 

Hence, [y] — 0, is a minimum, and [?/U = — , is a 

3 ' 3 Zl 

maximum. 



80 DIFFERENTIAL (JALUULUS. 

If y represents any function of x> and if u = y n , we 

have, 

du «_i du 

— = ny n 1 -f. 
ax u ax 



If x = a reduces -j- to 0, but does not reduce y n 1 to U, 

ax 

we have, from what precedes, 



If w?/ w_1 is +, any value that makes y a maximum, or 
minimum, also makes w a maximum, or minimum; if 
n?/^" 1 is — , any value that makes y a maximum, or mmi- 
m^/K, makes w a minimum, or maximum. The converse 

is also true, if we except the values that make y n equal 
to 0. Hence, if care be taken to reject the exceptional 
values, we may throw off a radical sign in seeking for a 
maximum or minimum. 



5. Let y — V±a*x 2 — 2ax*. 

Throwing off the radical sign, suppressing the fact t, 
+ 2a, and denoting the result by u, we have, 

-j- = lax — 3x 2 = ; .\ x = 0, x = — . 
dx 3 



The value, x = 0, makes y — 0, and is to be rejected. 
The value x =z ±a makes -j- % negative, and gives 

y = — , a maximum, 

3 V3 



MAXIMA AXD MINIMA. 81 

In like manner, if we have u — ly, we have, 

— - = — . -¥-. and if \ ~r \ -=-- 0, we have, as before, 
dx y dx LdxJ a 



Ldx*J a Ly ' dx*J a ' 



Hence, we infer, as before, that we may, with proper pre- 
caution, treat the logarithm of a function instead of the 
function itself, and the reverse 

, T , x* -3x+ 2 ( x -i)( x -2) 

6. Let y = — _, or v — -j -{) -f. 

J x* + dx + 2' •' (x + l){x + 2) 

Passing to logarithms, and denoting ly by u, we have, 

u = l(x - 1) + l(x - 2) - 7(a: + 1) - Z(a + 2). 



Whence, 



du _ 1 1 1 _ J_ _ 

flfo; 2; — 1 a; — 2 x + \ x + 2~ 



6(^ 2 - 2) , , ,- 



Both values of £ make ?/ negative. The first makes 

TO 

-7— „ negative, and the second makes it positive. Hence we 
dx 2 

have, y = 12 V2 — 17, minimum, and y = — 12 V 2 — 17, 
maximum. 

PROBLEMS IN MAXIMA AND MINIMA. 

1. Divide 21 into two parts, such that the less multiplied 
by the square of the greater, shall be a maximum. 

Solution. — Let x be the greater, and 21— x the less. We 
4* 



8k DIFFERENTIAL CALCULUS. 

have, for the equation of the problem, y — (21 — x)x 2 . 
By the rule we find that x = 14 makes y a maximum 
Hence, the parts are 14 and 7. 
. 2. Find a cylinder whose total surface is equal to 8, and 
whose volume is a maximum. 

Solution. — Denote the radius of the base by x, the alti- 
tude by y, and the volume by V. From the geometrical 
relations of the parts, we have, 

8 = 2*x 2 + 2«x X y, and V = *x* X y. 
Combining, we have, for the equation of the problem, 

^ a 8 - 2*x 2 1 , '. x 

F=«»- ns5 --=j. (ft -*«»). 

Fis a maximum when £ = i / — ; whence, y = 2i/ — , 

or y — 2x. That is, the altitude is equal to twice the 
radius of the base. 

3. Find the maximum rectangle that can be inscribed 
in an acute-angled triangle whose base is 14 feet, and alti- 
tude 10 feet. 

Ans. The base is 7 feet, and the altitude 5 feet. 

4. Find the maximum triangle that can be constructed 
on a given base, and having a given perimeter. 

Solution. — Denote the base by b, a second side by x, the 
third side by 2p — b — x, the perimeter by 2p, and the 
area by A. From the formula for the area in terms of 
the three sides, we find the equation of the problem, 



A = Vp(p — b) (p — x) (b + x — p). 

For x = p — %b, A is a maximum. Hence, the triangle is 
isosceles. 



MAXIMA AND MINIMA. 83 

5. To find the maximum isosceles triangle that can be 

inscribed in a circle. 

Ans. An equilateral triangle. 

6. To find the minimum isosceles triangle that can be 
circumscribed about a circle. 

Ans. An equilateral triangle. 

7. To find the maximum cone that can be cut from a 
sphere whose radius is r. 

0,^,4/2 4 

Ans. The radius of the base is — = — , and the altitude is -v. 

o o 

8. To find the maximum cylinder that can be cut from 
a cone whose altitude is h, and the radius of whose base 
is r. 

Ans. The radius of its base is -r, and its altitude -h. 

o o 

0. To find the altitude of the maximum cylinder that 
can be cut from a sphere whose radius is r. 

Ans. Altitude = o^a/3 

10. To find the maximum rectangle that can be inscribed 
in an ellipse whose semi-axes are a and b. 

Ans. The base is «a/2, and the altitude b a/2. 

11. To find the maximum segment of a parabola that 
can be cut from a right cone whose altitude is h, and th^ 
radius of whose base is r. 



Ans. The axis is equal to jV^ 2 + r 2 

12. To find the maximum parabola that can be inscribed 
in a given isosceles triangle. 

Ans. The axis is equal to u " the altitude* 



84 DIFFERENTIAL CALCULUS. 

13. To find the maximum cone whose surface is cv a 

gtant, and equal to S. 

1/S 
Arts. The radius of the base is ^V ^' 

14. To find the maximum cylinder in at can be cut fro a 
h given oblate spheroid, whose semi-axes are a and b. 

Ans. The radius of the base = ay ,,, 

o 

2 

and the altitude = b— -=, 

Vs 

15. From the corners of a rectangle whose sides are a 
and b, four squares are cut and the edges turned up to form 
a rectangular box. Required the side of each square when 
the box holds a maximum quantity. 

Ans. °^- b - lVa* - ah +> 

b b 

16. To find the minimum parabola that will circum- 
scribe a given circle. 

Solution. — Let x and y be the co-ordinates of the point 
of contact, a and b the terminal co-ordinates, 2p the vary- 
ing parameter, and r the radius of the circle. For any 
circumscribed parabola, we have, 



a — x + p + r, and b = v'lpa = \%p{x + p + ?•) (1) 

But from the triar.gle, whose sides are the radius, the sub- 
normal, and the ordinate of the point of contact, we have, 

and b — p 4- r (2) 

It will be shown hereafter that the area of a parabola ie 
two-thirds the rectangle having the same base and altitude 



MAXIMA AND MINIMA. 85 

Assuming this property, and denoting the area by A, we 
have, for the equation of the problem, 

i-J. fc .,-JJe±d: (8 , 

Applying the rule, and remembering that p and A are 
variable, we find that A is a minimum when 2p = -. 

17. Find the maximum difference between the sine and 

versed-sine of a varying angle. 

Ans. When the arc is 45°. 

18. Find the maximum value of y in the equation 
y = x 1 - 1 *. 

Ans. y' = e 4 . 

19. Divide the number 36 into two factors such that the 
sum of their squares shall be a minimum. 

Ans. Each factor is equal to 6 

20. Divide a number m into such a number of equal 
parts that their continued product shall be a maximum. 

m 

Solution. — Let x denote the number of parts, — one 

x 

fm\ x 
part, and ( J the continued product ; hence, the equation 

of the problem is, 

v = ( ) ; or, ly = xl — = xlm — xlx. 

\X J X 

J- — y(lm - 1 — Ix) = ; .*. Ix = Im — \—lm — le - I— A 

\ x = —, and y = 



86 DIFFERENTIAL CALCULI! b. 



r(Py_ 
ldx 2 J' m 



]«H> x -il- = -*\ x .s ; ■•■*= 



is a maximum. 

21. What value of x will make the expression 

1 4- tana; 

a maximum ? 

Ans. x = 45°. 

22. Find the fraction that exceeds its square by the 
greatest possible quantity. 

A 1 

Ans. . 

2 

23. The illuminating power of a ray of light falling 
obliquely on a plane varies inversely as the square of the 
distance from the source, and directly as the sine of its 
inclination to the plane. 

How far above the centre of a horizontal circle must a 
light be placed that the illumination of the circumference 

may be a maximum ? , r 

J Ans. 4- — -, 

Maxima and Minima of a Function of Two Variables. 

49. A function of two variables is at its maximum state 
when it is greater, and at its minimum state when it is less 
than all its consecutive states. 

If two parallel sections (Art. 23) be taken through a maxi- 
mum or minimum ordinate of a surface, the ordinate will be a 
maximum or minimum in each section, and in nearly 
every practical case the reverse will hold true. The ex- 
ceptional cases will bb these in which the maximum or 
minimum ordinate corresponds to a singular point of the 
surface. Omitting these, the problem is reduced to find 
tiig an ordinate that shall be a maximum, or a minimum 



MAXIMA AND MINIMA. 87 

in both the parallel sections through it This requires 

dz dz 

that — and — be simultaneously equal to 0. The equa- 

cLy dx 

tions thus obtained will determine all the values of x and 
y that correspond to maxima or minima states, and each 
set of such values may be tested in the manner explained 
in the last article, observing that for the section parallel 
to the plane xz, the successive differential coefficients of z } 
are found by supposing y constant, and for the section 
parallel to yz, they are found by supposing x constant. 









EXAMPLES. 








1. 

We 


Let z ■■ 
have, 


= x 3 


— 3xy + y 3 . 










dz 
dx 


--3x 2 


— 3y = 0, and -7- = 
y dy 


3y 3 - 


-3x 


= 






.'. X - 


= 0, y = ; and x = 


i,y-- 


= 1. 





AisOj 

d 2 z „ d*z , , d 2 z n d*z n 

The first set of values reduce the second differential 
coefficients to 0, but not the third differential coefficients; 
hence, they are to be rejected. The second set of values 
make both of the second differential coefficients positive; 
hence, they correspond to a minimum, which is z = — 1. 

2. Let z — x s y 2 — x*y 2 — x z y z . 

1 1 1 

For, x = -, y — -, z— — , a maximum. 



3. Let z — Vp(p — x)(p — y)(x-\-y — p). 
Dropping the radical sign, passing to logarithms, and 
denoting the logarithm of z 2 by u, we have, 

u = l(p) + I(p - x) + l(p -y) +l(x + y - p). 



88 


DIFFERENTIAL 


CAI 


Hence, 






du 


1 1 

— + 




dx 


p — x x + y - 


-p 


du 
dy~ 


1 + * 

p-y % + y- 


-p 



0; 



0, 



2 2 

x = qP' y = a P> ma, ke 2 a maxim um. 
o o 



V. Singular Values of Functions. 

Definition and Method of Evaluation. 

50. A singular value of a function is one that appears 
under an indeterminate form, for a particular value of the 

variable. Thus, the expression -— reduces to -, for 

1 x 3 

the particular value, x = 0, whereas its real value is -. 

6 

Singular values that take the above form of indeter- 

mination, may often be detected, and their real value 

found by means of the calculus. Let u = -. in which z 

y 

and y are functions of x that reduce to for x — a. Clear- 
ing of fractions and differentiating, wc have, 

udy + ydu — dz ; 

if we make x — a, the second term disappears, and we have, 

dz- 

dy. 

If both dz and dy reduce to for x = a, we have, in like 

manner, 

and so on. Hence, the following rule: 



M«=[jvl- 



SINGULAR VALUES OF FUNCTIONS. 81? 

Divide the successive differentials of the numerator by the 
corresponding differentials of the denominator ; substitute 
the particular value of the variable in the resulting fraction, 
continuing the operation till one is reached that is not inde- 
terminate ; the value thus found is the value required. 

Thus, in the example above, we have, 

rx — sin:tn rl — coson _ rsinari _ rcosari 1 
L x~* Jo ~~ L 3x 2 Jo ~ lOx Jo ~~ L - 6~ Jo ~ 6' 



EXAMPLES. 

1. Find the value of = . Ans. z 

L x 3 — 1 Ji 

r- %2 2X ~\ 1 

55. Find the value of — s — — . Ans. «. 

Lc 4 — 2a; 3 + 8x - I6J2 8 

„„.-,,, , n rx 3 — a 2 x — ax 2 + a s ~\ . n 

8. Find the value of 5 . Ans. 0. 

L x 2 — a 2 la 

rl - % n l 

4. Find the value of ., . Ans. n. 

L 1 — x Ji 

5. Find the value of P* ~ * x l • ^ s - 2 

LZ(1 + x) Jo 



Z(l + x) 
1 — cos 



/l?^. 0. 



6. Find the value of f= — S -^| . 

L x Jo 

7. 7ind the value of [ g , + ^,1*, _ a ] x • -' «■ * 



8. Find the value of [p^i] ■ Afu ' 2 



itf) DIFFERENTIAL CALCULUS. 



9. Find the value of 1 Ans. 

LI -f tsMXJrt 



4 



10. Find the value of ["- w * x a 1 • Ans. 2, 

*-2x - V5z 2 - a zJa 

Singular values that appear under the forms X », 
— , and oo — oo , can be reduced to the form discussed, and 

X) ' ' 

then treated by the rule. The method of proceeding in 
each case will be illustrated by an example. 

11. The expression (1 —x) tan -^-, which reduces to x oo , 

<i 

1 — x 
when x = 1, can be placed under the form , whieh, 

cot -x 
</ 
2 
by the rule, reduces to - for the particular value, x = 1, 



i(X x 2 

12. The expression, tan— -f- t-— — , which reduces to 

/j [X I ) 

X 2 — 1 

^- for z = 1, can be placed under the form , and 

z 2 cot-a: 
Z 

4 
this, by the rule, gives for x = 1, the value . 

if 

if 

1 3. The expression, xtanx — ^ secz, which reduces to 

. ,. . * , • <u a^ina* - 1* 
on — oo , for the value z = -, may be written =-, 

' 2 J cosz 

which, by the rule, gives for x = ^, the value — 1. 



ELEMENTS OF GEOMETKICAL MAGNITUDES. 



91 



VI. Elements of Geometrical Magnitudes. 

Differentials of Lines, Surfaces, and Volumes. 

51. Let AP and BQ be two consecutive orumates of the 
otirve, KL, whose equation is y = f(x); then will PQ bt 
the differential of the length of the 
curve, and AP QB will be the dif- 
ferential of the area, bounded by 
the curve, the axis of x, and any 
two ordinates; if we suppose the 
figure to revolve about the axis of 
x, the curve will generate a surface 
of revolution, and the area between 
the curve and axis will generate a 
volume of revolution; the surface generated by PQ, is the 
differential of the surface of revolution, and the volume 
generated by APBQ, is the differential of the volume of 
revolution. When the equation of XL is given, the values 
of these differentials may always be found in terms of a 
and dx, or of y and dy. 

1°. Denote PQ by dL ; we shall have, as in Art. 5, 




Fig. n. 



dL = Vdx 2 + dy 2 



(1) 



To apply this formula, we differentiate the equation ol 
the curve, combine the resulting equation with that of the 
curve, so as to find the differential of one variable in terms 
of the other variable and its differential, and substitute 
this in the formula. Thus, let it be required to find the 
differential of the arc of a parabola. Assuming the equa- 
tion of tLc parabola, y 2 = 2px, we find, by differentia tioE 
and combination, 



^ DIFFERENTIAL CALCULI. 

dy = dxi/ JL, and dx = %-dy, or dy 2 ~- dx*£ 

v 2 
and dx 2 = ~dy 2 . 
p 2 * 

Substituting, and reducing, we have, 

dL = tfej/^A or dL = ^VPTJ'. 

'Z°. Denote APQB by dA ; this is made up of two parts, 

the rectangle, AR = ydx, and the triangle, RPQ = -dydx 3 

but the latter is an infinitesimal of the second order, it may 
be therefore neglected in comparison with the former; 
hence, we have, 

dA = ydx (2) 

This formula may be applied in a manner similar to that 
just explained. Thus, if it be required to find the differ- 
ential of the area of a parabola, we have, 

V 2 / — 

dA = —dy ; or, dA — (v'Zpx)dx. 

3°. Denote the surface generated by PQ by dS ; this 
surface is that of a frustum of a cone, in which the radius 
of the upper base is y, the radius of the lower base, y + dy % 
and the slant height, ydx 2 + dy 2 . Hence, 

dS = \[^y + M# + dy)] X Vdx 2 + dy 2 . 

Neglecting dy in comparison with y, and reducing, we 
have, 



dS=Z*t/\d.r* +dy 2 (3) 



APPLICATION TO POLAE CO-ORDINATES. 



93 



To find the differential of a paraboloid, we proceed as' 
before. Substituting the values already found, and re- 
ducing, we have, 

2*ydy 



dS = 2«dxV2p^+ p 2 ; or, dS = ^M^L Vp 2 + y 2 - 

P 

4°. Denote the volume generated by APQB, by dV / 
this volume is that of a frustum of a cone, in which the 
radius of the upper base is y, of the lower base, y + dy, 
and the altitude doc. Hence, 



dV 



[Vt/ 8 + ir(y + dy) 2 -f ny(y + dy)]dx 



neglecting dy in comparison with y, we have, 

dV~ «ij*dx (4) 

Applying the formula to the paraboloid of revolution, wc 
have, as before, 

dV= ir — dy, or, d V = 2*pxdx. 



VII. Application to Polar Co-ordinates. 

General Notions, and Definitions. 

52. In a polar system, the 
radius vector is usually taken 
as the function, the angle be- 
ing the independent variable. 
Let P be any point of the 
curve, PL, in the plane of 
the axes, OX, OY; let 
be the pole, and OX the ini- 
tial line, of a system of polar 
co-ordinates. Then will OP, 




94 DIFFERENTIAL CALCULUS. 

denoted by r, and. XOP, denoted by 9, be the polar co 
ordinates of P, and the equation of PL may be written 
under the form, 

r =/(<>). 

It will be convenient to express the values of <p in terms 
of ir as a unit ; in this case, it is laid off on the directing 
circle, GS, in the direction of the arrow when positive, and 
in a contrary direction when negative. Let GH be the 
measure of <p for the point P, and let HI be equal to the 
constant infinitesimal, dy ; draw OIQ, and with as a 
centre describe the arc, PB ; then will OP and OQ be 
consecutive radius vectors, P and Q will be consecutive 
points, RQ will be the differential of r, PQ will be the 
differential of the arc, POQ will be the differential of the 
area swept over by the radius vector, and PQ prolonged, 
will be tangent to the curve at P. 

The line, BC, perpendicular to the radius vector, OP, is 
the movable axis, and the perpendicular distance, OA, is the 
polar distance of the tangent. Prolong the tangent to meet 
the movable axis at B ; at P draw a normal, and produce 
it to meet the movable axis at C; then is OB the subtan- 
gent, PB the tangent, OC the subnormal, and PC tht 
normal at the point P. 



Useful Formulas. 

53. 1°. Differential of the arc. Denote the differential 
of the arc, PQ, by dL ; RP being infinitesimal may be re- 
garded as a straight line perpendicular to OQ ; it is equal 
to rdv : hence, 



dL = Vdr* + r s <fo* (1 



APPLICATION TO POLAR CO-ORDINATES. 95 

2°. Differential of the area. Denote the differential of 
the area swept over by the radius vector by dA ; this is 
made up of the sector, OPR, and the triangle, RPQ ; 
but, RPQ is an infinitesimal of the second order, and the 
sector is an infinitesimal of the first order ; neglecting the 
former in comparison with the latter, and remembering that 
the area of a sector is equal to half the product of its arc 
and radius, we haye, 

dA = lr*dtp (2) 

3°. Angle between the radius vector and tangent. Denote 
the required angle by V; the angle, RQP, differs from the 
required angle, OPB, by an infinitesimal, and may, there- 
fore, be taken for it; but tan RQP, equals RP, divided by 
RQ ; hence, we have, 

*»r = % (3) 

4°. Polar distance of the tangent. Denote the distance 
OA by p ; the triangles, QPR and Q OA, are similar : 
hence, 

QP : RP : : QO : OA. 

But, PO differs from QO by an infinitesimal, and may. 
therefore, be taken for it ; making this change, and sub' 
stituting for the quantities their values, we have, 



hence, 



Vdr 2 + r 2 d$ 2 : rdcp :: r : p; 
r 2 d(p 



Vdr 2 +r 2 d(D % 



w 



m 



DIFFERENTIAL CALCULUS. 



5°. Formula for subtangent. Denote the subtangent b^ 
S.T; in the triangle, POP, the perpendicular, OP, ig 
equal to the base, OP, multiplied by the tangent of the 
angle at the base; hence, 



S.T 



~dr~~ 



(5) 



6°. Formula for subnormal. Denote the subnormal by 
S.N; the triangles, OPB and OOP, are similar; hence, 
but, OC, equals OP, divided by 



tsmOCP = ttmOPB; 
tsmOCP; hence, 



s.ir = 



dr 

dq> 



(6) 



7°. Formula for the radius of curvature. Denote the 
radius of curvature by. p; let P and Q be two consecutive 
points of the curve, XL, and let 
PC and QO be normals at these 
points, meeting at C ; then will 
C be the centre of the oscillatory 
circle ; draw OM perpendicular 
to PC, it will be parallel to the 
tangent at P, and, consequently, 
the intercept, PJSI, will be equal 
to the polar distance of the tan- 
gent denoted by p ; draw C, 
OP, and Q. From the figure, we have, 




Fi£. 13. 



OC 2 = r* -f- p 2 -2p P . 

If we pass from P to Q, r will become r + dr, p will 
become jo 4- dp, and OC and p will remain unchanged. 
Lf, therefore, we differentiate the preceding equation under 
the supposition that r and p are variable, the resulting 



APPLICATION TO POLAR CO-ORDINATES. 9? 

equation will express a relation between p,, r, and p. Dif- 
ferentiating, we have, 

2rdr - if dp = 0; .-. , = ±- (7) 

8°. Formula for the chord of curvature. The chord ol 
curvature is the chord of the osculatory circle that passes 
through the pole and the point of osculation. Denote it 
by 0; prolong PC and PO till they meet the circle at R 
and N, and draw RN. The right angled triangles, PNR 
and PMO, are similar, hence, 

PJST : PR : : PM : PO, 

or, 

: 2p : : p : r ; 

nence, 

"=£«* = *$ (8) 



Spirals. 

54. If a straight line revolve uniformly about one of its 
points as a centre, and if, at the same time, a second point 
travel along the line, in accordance with any law, the lat- 
ter point will describe a spiral. The part described in one 
revolution of the straight line is a spire. The fixed point 
is the pole. If we denote the distance from the pole to the 
generating point by r, and the corresponding angle, 
counted from the initial line, by <p, we have, for the gen- 
eral equation of spirals, 

r -/(?) (1) 

When this relation is algebraic, the spiral is said to be 
algebraic; when this relation is transcendental, the spiral 
is said to be transcendental. 



^8 DIFFERENTIAL CAL.CUJLUB. 

Among algebraic spirals, the most important are the 
spiral of Archimedes, the parabolic spiral, and the hyper- 
bolic spiral, corresponding, respectively, to the right line> 
the parabola, and the hyperbola. 

Spiral of Archimedes. 

55. The equation of this curve is, 

r = ay (2) 

We see that the generating point is at the pole when 
the variable angle is 0, and that the radius vector increases 
uniformly with the variable angle, being always equal to o 
times the arc of the directing circle that has been swept 
over. Differentiating equation (2), we have, 

dr = ady. 

Substituting, in formulas (1) to (8), Art. 53, we find, 



dL = adyVl + 9 2 ; &T = ay 2 ; 
dA = -z-a 2 cp 2 d(p; S.JST=a; 



TanF 



a(l + y 2 )$ 



a' r 2 + 9 2 

_ acp 2 § _ 2atp(l + <p«) ; 

P ~VlTv* : 2 + 9 2 ' 

If we take a straight line, whose equation is y = ax, and 
lay off the abscissa of any point on the directing circle, 
and the ordinate of that point on the corresponding 
radius vector, the point thus determined is a point of the 
spiral of Archimedes. 



TRANSCENDENTAL CURVES. 



99 



In like manner, we may discuss and construct the para- 
bc'ic spiral, whose equation is r 2 = 2p$, and the hyperbolic 
spiral, whose equation is r<p = m. The former corresponds 
to the ordinary parabola, and the latter, to the ordinary 
hyperbola referred to its asymptotes. 



VIII. Transcendental Curves. 

Definition. 

56. A transcendental curve, is a curve whose equation 
can only be expressed by the aid of transcendental quanti- 
ties. The cycloid, and the logarithmic curve are examples 
of this class of lines. 

The Cycloid. 

57. The cycloid is a curve that may be generated by a 
point in the circumference of a circle, when that circle 
rolls along a straight line. The point is called the genera- 
trix, the circle is called the generating circle, and the 
straight line is the base of the cycloid. The curve has an 
infinite number of branches, each corresponding to one 
revolution of the generating circle. 

To find the equation of one branch, A PM ; let A be tn« 
origin, the centre of the generating circle, and P the 
place of the generatrix after the circle has rolled through 
the angle KGP, denot- 
ed by <p ; denote the co- 
ordinates, AL and LP, 
by x and y, and let the 
radius of the generating 
oi rcle be represen ted by r. 




AL 



Fig. 14. 



From the figure, we have, 



AL = AK-LK. 



100 DIFFERENTIAL CALCULUS. 

But, AK is equal to the arc KP, and KP is the are 
whose versed sine is y, to the radius r, or r times the arc 

whose versed sine is — , to the radius 1 ; LK is equal to 

PD, which, from a property of the circle, is equal to 
=fc s\/(2r — y)y, the upper sign corresponding to the case 
in which P is to the left of KE, and the lower sign to the 
case in which P is to the right of KE ; and AL is equal 
to x. Substituting these values, and reducing, we have, 

x = rversin - —^ =p \/%ry — y 2 ; 

which is the equation sought. From it we see that y can 
never be less than 0, nor greater than 2r; we see, also, that 
y is equal to 0, for x — 0, and for x = 2*r, and that for 
every value of y, there are two values of x, one exceeding 
irr as much as the other falls short of it. 
"Differentiating and reducing, we have, 

V%ry — y % 

which is the differential equation of the cycloid. From it 
we deduce the equation, 



i/j- 



r 

dx~ ~y y *' ' dx*^ - ^* 



From the first, we see that the tangent to the curve is 
vertical for x = 0, and x = 2*r, and that it is horizontal 
for x = tr ; from the second, we see that the curve is always 
concave downward. 

Substituting, in formulas (6) and (7), Art. 32, we have. 

&T=±—J— , and S.N=+V%ry"-v"\ 

^/%ry" — y"* * 



TKAKSCEKDEtfTAL CURVES. 



101 



The Logarithmic Curve. 

58. The logarithmic curve, is a curve in which any 
ordinate is equal to the logarithm of the corresponding 
abscissa. Its equation is, therefore, 

y = logic, or y = M . Ix; 

in which M is the modulus of the 
system. 

If M > 0, y will be negative when 
x < 1, positive when x > 1, when 
x = 1, and infinite when x = 0. 

If M < 0, y will be positive when 
c < 1, negative when x > 1, when x 
when x = 0. 

In no case can a negative value of x correspond to a 
point of the curve. 

Differentiating, we find, 

dy _ M d 2 y _ M 

dx x 1 dx 2 x 2 ' 




Fig. 15. 



1, and infinite 



The first expression has the same sign as 31, and varies 
inversely as x. Hence, when My 0, the slope is positive, 
when M < 0, the slope is negative, and in both cases the 
curve continually approaches parallelism with the axis 
of x. 

The second expression has a sign contrary to that of M, 
and varies inversely as the square of x. Hence, when M > 0, 
the curve is concave downward, as KDC, and when M < 
it is concave upward, as KDC. 



PART III. 

INTEGEAL CALCULUS 



Object of the Integral Calculus. 

59. 'dhe object of the integral calculus is to pass from a 
given differential to a function from which it may have 
been derived. This function is called the integral of the 
differential, and the operation of finding it is called inte- 
gration. The operation of integration is indicated by this 

sign, /, called the integral sign. Integration and differ 

entiation are inverse operations, and their signs, when 
placed before a quantity, neutralize each other; thus, 



/ 



dx = x. 

A constant quantity connected with a function by the sign 
of addition, or subtraction, disappears by differentiation , 
hence, we must add a constant to the integral obtained ; 
and inasmuch as thii constant may have any value, it is 
said to be arbitrary. The integral, before the addition of 
the constant, is called incomplete, after its addition it is 
said to be complete. By means of the arbitrary constant, 
as we shall see hereafter, the integral may be made to 
satisfy any one reasonable condition. 

Nature of an Integral. 

60. The differential of a function is the difference 
between two consecutive states of that function ; hence. 



NATURE OF AX INTEGRAL. 103 

any state of the function whatever, is equal to some pre- 
vious, or initial state, plus the algebraic sum of all the 
intermediate values of the differential. But this state is 
the integral, by definition ; hence, if the arbitrary constant 
-epresents the initial state, the incomplete integral repre- 
jents the algebraic sum of all the differentials from the 
initial state up to any state whatever, and the complete 
integral represents the initial state, plus the algebraic sum 
of all the differentials from that state up to any state 
whatever. 

Before any value has been assigned to the constant, the 
integral is said to be indefinite; when the value of the 
constant has been determined, so as to satisfy a particular 
hypothesis, the integral is said to be particular ; and when 
a definite value has been given to the variable, this integral 
is said to be definite. 

The values of the variable that correspond to the initial 
and terminal states of a definite integral are called limits, 
the former being the inferior, and the latter the superior 
limit ; the integral is said to lie between these limits. The 
value of the definite integral may be found from the indefi- 
nite integral by substituting for the variable the inferior 
and superior limits separately, and then taking the first 
result from the second. The first result is the integral 
from any state up to the first limit, and the second is the 
integral from the same state up to the last limit; hence, 
the difference is the integral between the limits. The 

symbol for integrating between the limits is / » m which a 

O a 

is the inferior, and b the superior limit. 

This article will be better understood when we come tc 

its practical application in Articles 78 and following. 



104 INTEUitAL CALCULUS. 



Methods of Integration ; Simplifications. 

61. The methods of integration are not founded on pro- 
cesses of direct reasoning, but are mostly dependent on 
formulas. The more elementary formulas are deduced by 
reversing corresponding formulas in the differential cal- 
culus ; the more complex ones are deduced from these by 
various transformations and devices, whereby the expres- 
sions to be integrated are brought under some known 
integrable form. 

It has been shown (Art. 9), that a constant factor re- 
mains unchanged by differentiation; hence, a constant 
factor may be placed without the sign of integration. It 
was also shown in the same article, that the differential of 
the sum of any number of functions is equal to the sum 
of their differentials; hence, the integral of the sum of 
any number of differentials is equal to the sum of theii 
integrals. These principles are of continued use in inte- 
gration. 

Fundamental Formulas. 

ax n + 1 

62. If we differentiate the expression -, we have, 

n + 1 • 

„tt + l 



,fax T \ n 

d[ - = ax ax; 



reversing the formula, and applying the integral sign tc 
Hoth members, 



Remembering that the signs of integration and differentia- 



METHODS OF INTEGRATION. 105 

tion neutralize each other (Art. 59), and adding a constant 
to complete the integral, we have, 

faz n dx = —^- + G [1] 

J n + 1 

Hence, to integrate a monomial differential, — drop the 
differential of the variable, add 1 to the exponent of the 
variable, divide the result by the new exponent, and add a 
constant 

This rule fails when n = — 1 ; for, if we apply it, we get 
a result equal to qo ; in this case the integration may be 
effected as follows : 

From Art. 15, we have, 

d(alx) = a — = ax'~ 1 dx; 
x 

reversing and proceeding as before, we have, 

/' _i fdx 

ax dx = af — = alx + G • • . . . . j 2 J 

From Art. 16, we have, 



ii) 



a x dx ; 
reversing and proceeding as before, we have, 



/ 



r z = r + G [3J 



In like manner, reversing the formulas in Article 17. let 
tered from a to h, we have, 



/• 



cosxdx = sins + C |4 

8* 



106 INTEGRAL CALCULUS. 

/ — siaxdx = cosrr + C [5] 

/*-%- = tana: + 6' [6] 
cos 2 z l J 

r jx = Gotx c 

J sin 2 z l J 

I sinxdx = versing + G [8 J 

I — cosxdx = coversinz + [9J 

I t&nxsecxdx = seat + [10 1 

1— cotzcoseczdz = cosec^ + C [ 1 1 J 

[ti like manner, from the formulas in Art. 18, lettered from 
i' to h', we deduce the following: 

f-v£* mm **' +a |131 

/rlF = bm " ly + c7 fl4) 

/-rT7 = 00t " ly + [15] 



METHODS OF INTEGRATION. 107 



/' =M= = versin-V + [16J 

~ — = coversin -1 2/ + C. . . [17] 

V2y -y 2 

C— P— = sec" V 4- C [18 j 

J 2/Vy 2 — 1 

- p- — - = cosec- V + O [19J 

yvy 2 — 1 



If in formulas (12), (13), (14), (15), (18), (19), we make 

bx , , bdx , , . 

V = — , whence ay = — , and reduce, we hare, 



/dx 1 . _i bx ~ rArt , 

— = rsin x — + .... [20J 
Va 2 - b 2 x 2 b a 

; = t COS — + C. . . f21 

a/« 2 - 6 2 z 2 * a 

rf*L tan -ite + paj 

J a 2 + & 2 :e 2 ab a L J 

/- a -^w = r5 cot 7 + c ••■•[»] 



— = - sec — + G. . . . 1 24 j 

xV¥x^- a 2 a a 



108 INTEGKAL CALCULUS* 



/ 



dx 1 _i bx n _ 

= - cosec 1- C7 . . . . [2oJ 



x^/i,2 x 2 _ a z a a 



2b 9 
From (16) and (17), by substituting — x for y, and re 

ducing, we have, 



/ 



dx 1 . _i 2b 2 FO _. 

= ^ versin — =- x + C [26J 



Va 2 x — b 2 x 2 b 



f- 



dx 1 . _i 2b 2 „ r _ wi 

= - coversm — - x + C . . [27J 



Va 2 x - b 2 x 2 b a '' 



Formulas (1) to (19) are the elementary forms, to one 
of which we endeavor to reduce every case of integration ; 
the processes of the integral calculus are little else than a 
succession of transformations and devices, by which this 
reduction is affected. 

In the following examples, and, as a general thing, 
throughout the book, the incomplete integral is given, it 
being understood that an arbitrary constant is to be added 
in every case. 

EXAMPLES. 

Formulas (1) to (3). 
I. dy — 12 3 dx. Arts, y — - 



l 2 1 

2. dy — .X s dx. An*, y — - b.> * 



3. dy — 3x A dx. Jnx. y — — x 



METHODS OF INTEGRATION. 



4. dy = Ix^dx. 

2 _£ 

5. dy = — -x dx. 

6. dy = (—ax s — —lx^\dx. 



S. 



= — x dx. 
-1 



9. % = 3x 1 dx. 

10. ^= (2^ + z _1 )efc- 

11. dy = e 8mx cosxdx. 

12. dy—— e cos *sinzdz. 



109 

7 i^ 
Ans. y zrz-x 1 . 

, 10 -i 

6 3 

,4m«. y = ax* —bx 2 . 



12 1 

Ans. y = + — . 

xx 3 

Ans. y = x*. 



Ans. y — dlx. 

6 5 
Ans. y = —x 3 + Ix. 
5 



Ans. y = e' 



,sm# 



^4^5. y — p. 



,co&r 



Formulas (4) to (11). 

13. dy = 2cos2xdx. Ans. y = sin22. 

14. dy = — — sm(# 2 ) X xdx. Ans. y = -t-cos(£ 2 ) 



15. dy 

16. dy 



dx 



cos 2 (£#y 

xdx 



sin 2 (3s»)' 



^4ws. y = 2tanf — :cj 



^ws. y = — cot(3T 2 ) 



110 

17. dy — sin(ax)dx. 



INTEGRAL CALCCJLUS. 



Ans. y = —versm(ax) 



18. dy = — cosf — x 2 j X xdx. Ans. y= coversinj — x 2 ) 



19. dy = 

20. dy = 



Formulas (12) to (19). 
2dx 



VI- 4z 2 
2xdx 



Ans. y — sin (2a;) 



-L 



VT 



Ans. y = sin (x 2 ) 



«• d V = 7Af=7r? 



xidx i d(x^/2x) 



2 - 4:x 3 3 Vl - 2x*' 



1 . 



22. Jy -- 



dx 



Ans. y = ^sin l (xV2x). 



9,Jh 



d(2x*) 



Vx - ix* ' Vl — ±x 



-l 



33 /y 



^ ma y — COS (2 V X) 



Xdx 1 </(.r 2 ) . 1 _i 

1 -I- r 4 2 1 -I- ^ 4 " 2 



:, fifa/ = — 



2dx 



35. </# 



1 +s*' 

dx 
x 6x — 9x* 



Ans. y = 2 cot a. 
.4ws. ^ = — versin — 1 3* 



METHODS OF INTEGRATION. Ill 



Formulas (20) to (25). 

26. dy = —J^=- ; (a = V5, b = 3). 

, 1 . -i 3a: 

4WS. W =-8111 = 

— — zcfo; 

27. ^ = ; (a= V2, 6= V~o, u = x*). 

V% - 5z 4 



1 -i^VS 

^ V5 V2 



28 ' ^ = irk*'' (a = 2,b = 3). 



, 1 _i 3* 

ilw*. # = —tan — 

* 2 2 



2//r 



-j gr 

A ns. y == cot — 



3a ^ = ,vfcl' («=V5,^V3). 



2 _irrV3 
<4tw?. u = — ^sec — 7= 

V5 V^ 



31. rft/ = - — ^L== ; (a = V5, b = VB). 
voa- 4 —2a; 2 

. 3 -ixV5 

^4/i.s. y ■= — - =cosec 



y/"i V% 



112 INTEGRAL CALCULUS. 



32. dy — - — — . Ans. y = — -=tan l xVB 

1 + 5a; 2 V5 



nn , - 2x~ 1 dx A 2 _i aVl4 

33. dy = — - — Ans. y = — -cosec — — - 

Vl4flJ 8 _ 3 a/3 V3 

Formulas (26) and (27). 

34. <fy = -J^= ; (fl = 2, b = 3). 

V4z 2 - $x* 

1 . _i9.r 8 
^aw5. y = — versm — — 
^3 2 



35. r/v = , ; (a = 1. 6 = V 2). 

V^b _ 2^ 

Ans. y = — -coversin~ 1 4.r 3 
3 2a/2 



EXAMPLES IN SUCCESSIVE INTEGRATION. 

36. d*y = ax dx z . 
Dividing by dx z , 

—¥- = axdx, or d[ — ~ ) = «2*&e. 
tu 2 \dx 2 / 

Integrating, we have, 

d 2 y ax* 

dx* 2 + J% 

Multiplying by dx, and integrating again, we find, 
dcf 6 



METHODS OF INTEGRATION. 118 

Multiplying again by dx, and integrating, 

37. d±y = - x~*dz*. 
A.S before, we have, 

dx*~ 3 + °' 



and final 



d*y x~ 2 , ^ „ 



dy x 1 Cx 2 __, 



1 7 Cx* Cx 2 , ,,„ 
y = 6 + ~6~ + "2" + + 



38. d 3 y - mdx*. 



i 
39. d 3 y =x?dx 3 . 



40. rf«?/ = ^ar'cfo*. 



mx 3 Ox' 
Ans. y — -£— + —jr- + C r z 4- f/ 



Ans. i/ — — -x* H h 6' a: + 6 

lOo 2 



Am. y — — — + fo + C 



il4 INTEGRAL CALCULUS. 

Integration by Parts. 
63. If u and v are functions of x, we have, from Art. 10 
d(uv) = udv + vdu, 
fntegrating and transposing, we have,. 

I udv = uv — I vdu [28] 

This is called the formula for integration by parts ; it 
enables us to integrate an expression of the form udv 
whenever we can integrate an expression of the form vdu, 
[t is much used in reducing integrals to known forms. 

EXAMPLES. 

1. dy — xlx dx ; 
Let Ix = u, and xdx = dv ; 

, dx , x* 

.*. du = — , and v = — . 
x 9 2 

Substituting in the formula, we have, 

_ x 2 Ix Pxdx _ x 2 lx x 2 
y ~~ ~2~ J T~ ~2 T 

^. <fy = - 



\&t yl — a; 2 = u, and — - = dv ; 

X s 

xdx . 1 

:, and v = — 



a/1-z 2 ' ^ 



METHODS OF INTEGRATION. 115 

hence, we have, from the formula, 



/_dx 



Vl — x* P dx Vl - 

y= z~ 



sm~*:& 

Vl 



Additional Formulas. 

64. 1°. Formula (1) may be extended to cover the case 
of a binomial differential of the form, 

dy = (a + lx n fx n - l dx i 

m which the exponent of the variable without the paren - 
thesis is 1 less than the exponent of the variable within. 

Let a + bx n = z ; .\ bnx n dx = dz, or x n dx = — . 

on 

Substituting, and integrating, we have, 



i=fia + i*r<r**=f% 



Pdz zP +1 



bn(p + 1) 
Replacing z by its equal, (a + bx n ), we have, 

/(. + h?V^* = ^fj^ + O [291 

Hence, to integrate a binomial differential of the proposed 
form, add 1 to the exponent of the parenthesis, divide tin 
result by the new exponent into the product of the coefficient 
and the exponent of the variable within the parenthesis. 

It is to be observed that a constant factor may be set 
aside during the process of integration, and then intro- 
duced, as explained in Article 61. 



116 INTEGRAL CALCULUS. 



EXAMPLES. 



1. dy = (l+ xrfxdx. Ans. y = (L±J?!!: 

xdx 



2. dy = :. ^ws. f/ = Vc 8 + a 2 

3. efy = 5* 3 dxVTT~3x*. Ans. y = ^-(7 + Sx*)* 

lo 

4. dy= (2 + 3x*) 3 xdx. Ans. y = ^- (2 + 3a; 2 )*. 
,, 7 2xdx . 1 

5. ^ = rr. ^TCS. y = — 



(*I + l) 2 * (*' + 1) 

2°. The preceding formula fails when p = — 1 ; in that 
case, we have, 

Keplacing 2: by its value, 

y*(a + ^-V- 1 ^ = }-l{ a + bx n ) + [30| 



EXAMPLES. 



1. dy = 3(3 -r 4z 3 )- 1 z 2 g^. vlw5. y .-= 1/(3 + 4z»), 

2. ^2/ = . Ans. y = l(x + a) 

x ~\- ct 

Ans. y = 2l(x + 4) 



x + 4 



METHODS OF INTEGRATION. 117 

When the differential expression is a fraction in which 
the numerator is equal to the differential of the denomi- 
nator multiplied by a constant, its integral is equal to that 
constant into the Napierian logarithm of the denominator. 

4 dy = sr=|t' Ans - y = - 3 ^ 3 - x3 "> 



3x* + 2x + 1 _ 
-dx. 



* X s + X 2 + X + 1 

^s. y = /(z 3 + # 2 + a* f 1 ) 
3°. Let it be required to integrate the expression. 

dx 

Vz 2 ± a 2 
assume, 

£ 2 =h a 2 = z 2 ; whence, xdx = zdz ; 

adding zdx to both members and factoring, ws have, 

(x 4- z)dx = z(dx + dz) ; 
hence, 

dx _ dx _ dx + dz _ d(x + z) > 
2 "~ y^ 2 ± a 2 ~ z + z "aTTT" ' 

applying the principle just deduced, we have, 

/. — - — = i{x + z) + a 
Vx 2 ±a 2 

Replaciug z by its value, we have, finally, 

f- ~^= = l(x + \/x^~±^) + C [31] 

J <Sx 2 ±a 2 



118 INTEGRAL CALCULUS 



!. dy = 



EXAMPLES. 

dx 1 dx 



V±x 2 -7 2 Vx 2 -1 



Ans. y = -l(x+ y^ g _ j 



3dx 
2. dy r= /l?is. y = 3Z(z + Vs*~^~5), 

Vx 2 — 5 • ' 

4°. Let it be required to integrate the expression, 

7 dx 

dy = -—=•; 

VSacc + a? 2 

since, dx = c?(a + a;), and 2ax + x 2 = (a + x) —a 9 
we have, 

dx p d(a + x) 



/dx p d(c 

V%ax + x 2 J Via + 



V(a + z) 2 - « 2 
which can be integrated by Formula (31); hence, 

— = l\a + x +V^z~+~x~ 2 I + G . . [34J 

V2rtfl: 4- z 2 i J 



1. dy = 



EXAMPLES. 

2dz cfo 



v/3z + 4z 2 V}^ + a 2 

3 



4*W. i/ = l\x + g + vfaT + » *) 



METHODS OF INTEGRATION. US 

3dx / 5 \ 

5°. Let it be required to integrate the expression, 

7 dx 

Factoring, we have, 

dx 1 l dx dx i 

a 2 — x 2 ~2a(a + x a — x) 

Integrating, we have, 

f^T* = i( l{a + x) - l{a - x) )- 

But the difference of the logarithms of two quantities is 
equal to the logarithm of their quotient; hence. 

and in like manner, 

x 2 -- a 2 2a x + a l 

EXAMPLES. 

, , dx . 1 . 3 + x 

_ , dx A 1 7 x — 2 

% dy = -- Ans. y = -I — — s 

* x* — 4 3 4 x + 9 



120 INTEGRAL CALCULUS. 

S. Given y-| = y, to find the relation between y and x 
Multiplying both members by 2dy, 

2dyd 2 y a , d(dy z ) n , 



fntegrating, 



,2 



= 2/ 2 + C; •*• dx 



_ ^ 



<&;• " ^2 _+_ # 

Integrating by Formula 31, 



x = Z(y + Vy 8 + C) + C". 

<# 2 S 

4. Given -rr— = — ?z 2 s, to find the relation between i 
at* 

and £, £ being the independent variable. 

Multiplying by 2ds, 

2dsd 2 s 



dt* 

In tegrating, 



2n 2 sds. 



C — n*s 2 ; .*. at 



dt* VC-n*s* 

Integrating by Formula (20), 

1 . _i ns „, 

t = - sin l — - + C. 



Rational and Entire Differentials. 

65. An algebraic function is rational and entire when rt 
contains no radical or fractional part that involves the 
variable. If the indicated operations be performed, every 



METHODS OF INTEGRATION. 121 

rational and entire differential will be reduced to the form 
of a monomial, or polynomial differential, each term of 
which can be integrated by formulas (1) or (2). Thus, 
if the indicated operations be performed, in the expression, 

we have, 

dy = - ( 8x* — ±x 5 + 6x* — SxAdx. 

Hence, by integration, 

1 / 8 , 2 « ■ 6 ■ 3 A \ 



-G 



i' , -k''*i,'-iy 



In like manner, any rational and entire differential may 
be integrated. 

Rational Fractions. 

66. A rational fraction, is a fraction whose terms are 
rational and entire. When a rational fraction is the differ- 
ential of a function, and its numerator is of a higher degree 
than its denominator, it may, by the process of division, be 
separated into two differentials, one of which is rational 
and entire and the other a rational fraction, in which the 
numerator is of a lower degree than the denominator. The 
former may be integrated as in the last article ; it remains 
to be shown that the latter can be integrated whenever 
the denominator can be resolved into binomial factors of 
the first degree with respect to the variable. The method 

a 



122 INTEGRAL CALCULUS. 

of integration consists in resolving the differential coeffi- 
cients into partial fractions, by some of the known methods, 
then multiplying each by the differential of the variable 
and integrating the polynomial result. There are four 
cases, depending on the method of resolving the differen- 
tial coefficient into partial fractions. Each case will be 
illustrated by examples. 

First Case. — When the binomial factors of the denomi- 
nator are unequal. 

Let us have, 

(2x — b)dx _ 2x — 5 

y ~ x* - 6x* + 113-6 ~ (x -l)(x- 2) (x - 3) 

Assume the identical equation, 

(1) 





2x 


-5 


A 

x — 


l + x 


B 

-2 + 


V 


(* 


-l)(x- 


- 2) (x - 3) ~ 


x-3 


1 


Clearing 


• of fractions. 










%x 


-5 = 


A(x-2)(x- 


3) + 


B(x- 


-1)0* 


-3) 








+ 


C(x- 


!)(•- 


-2) 



(2) 

In order to find A, B, and C, we might perform the 
operations indicated in (2), equate the coefficients of the 
like powers of x, and solve the resulting equations ; but 
there is a simpler method in cases like this, depending on 
the fact that (2) is true for all values of x. 

3 

Making x = 1, we find, — 3 = 2A; :. A = — -. 

Making x = 2, we find, — 1 = — B; .: B = 1. 
Making x = 3, we find, 1 = 2(7; /. C — x. 



METHODS OF IXTEGKATIOK. 123 

Substituting these in (1), multiplying by dx, and in- 
tegrating by Formula 30, we haye, 

/ (2x - b)dx _3 r dx P dx 

(x- l)(x-2)(x-3)~ 2 J x-1 + J x~^2 

+ \ fx^l = ~ 1 1{X ~ 1} + 1{X - 2) + 1 1{ - X ~ 3) * 

Reducing the result to its simplest form, in accordance 
with the elementary principles of logarithms, we have, 
finally, 

f x _ 2) (x- 3)4 
(x - 1)* 

EXAMPLES. 

, _ {& x + l)d x _ (& x + l)d x 
y ~ x*~+z-2 ~ (x-l){x + 2)' 

4ns. y = 2l(x - 1) + 3l(x +2) = lUx - l) 2 x (x + 2) 3 1. 
(x — l)dx (x — l)dx 



2. dy = 



x 2 + 6x + 8 (x + 2) (x + 4)' 



(x -f 4)^ 
^4ws. y = P '•—. 

(x + 2)* 

, ._ (2sc + 3)dz _ (2se + d)dx 
" y ~ x*+x 2 ~-^2x ~ x(x -1) (x + 2)' 

( x _ 1)3 

Ans. y =l— 3 '—p 

x*(x + 3)* 
_ (3x* - l)dx 

J ~~ x{x - 1 ) (x + 1)' 



124 INTEGRAL CALCULUS. 

Second Case. — When some of the binomial factors are 
equal. 

In this case there are as many partial fractions as 
there are binomial factors in the denominator; but the 
denominators of those corresponding to factors of tne 

form (x — a) m , are respectively of the form (x — a) m r 

(x — a) m ~ 1 , (x — a) m ~ 2 , etc., down to x — a. 

Let us assume, 

(x 2 + x)dx 



dy = 



(x- 2)» (x - 1)' 
assuming the identical equation, 



x 2 + x A BO 

+ " * + Z T (1) 



(x - 2) 2 (x - 1) (x - 2) 3 x — 2 x - 1 

and clearing of fractions, we have, 

x* + x = A{x- 1) +B(x-2)(x-l) + C(a?-2)»...(2) 

Here again we might find the values of A, B, a id C, by 
the ordinary method of indeterminate coefficients ; but it 
will be simpler to proceed as follows : 

Making x = 2, we find, A = 6 ; 

Making x = 1, we find, G = 2 ; 

giving to A and C their values in (2), and then making 
x = 0, we find, = - 6 + 2B + 8 .\ B = - 1. 
Substituting in (1) and multiplying by dx, we have, 

(x 2 + x)dx 6dx dx 2dx 



{x-2) 2 (x — l) {x-2) 2 x-2 x-l 



METHODS OF INTEGRATION. 12£ 

The first partial fraction can be integrated by Formula 
29, and the others by Formula 30; hence, 

y = ^ - l(x- 2) + 2l(x- 1) = - -^ + ¥- -f , 

y x — 2 v v a: — 2 £ — 2 



EXAMPLES. 

„ _ (x 2 — 2)dx 

In this example the equal binomial factors are of the 
form (x — 0) or x, and the assumed identical equation is, 

x 2 - 2 ABO D 

= z* + z* + -z + z—^ (l) 



X s (x— 1) X z X 2 X X 

Clearing of fractions, and performing indicated operations, 
we have, 

x 2 - 2 = Ax - A + #z 8 -Bx + Cx* - Cz 2 + Dx*. 

Equating the coefficients of like powers in the two mem- 
oers, and solying the resulting group, we have, A = 2, 
B = 2, (7 = 1, and D = — 1 ; substituting in (1), and 
proceeding as before, we have, 

2x + 1 . x 

y=--zzT- + iz-^i- 



x 2 dx 4 



3. cfy = 



(a + 2) (x + 3)2 

* + 3\ s 



3 ,/a; +3\ 

^ £ + 3 Vz + 2/ 



126 INTEGRAL CALCULUS. 

Third Case. — When some of the factors are imaginary 
but unequal. 

In this case the product of each pair of imaginary 
factors is of the form (x — a) 2 + I 2 ; instead of assuming 
a partial fraction for each imaginary factor, we assume 
for each pair of such factors a fraction of the form, 

A + Bx 

(x-a) 2 +b 2 ' 

Let dy = -. — w . 

* (aj + 1) (z 2 + 1) 

A.ssume the identical equation, 

x A + Bx G 

(x + 1) (x 2 + 1) ~" x 2 + 1 + a; + 1 * ' 

Clearing of fractions, and reducing, 

x = Bx 2 + (A + B)x+ A + Cx 2 + C (2) 

Equating the coefficients of the like powers of x in the twc 
members, and solving the resulting equations, we have, 

A=\, B = \, and G = -\. 

Subrtituting in (1), and multiplying by dx, we have, 

xdx _ 1 (1 + x)dx 1 dx 

Jx~i~T)(x 2 + 1) ~~ 2 cc 2 +1 2 cc + 1 

1 / dx xdx dx \ 

= 2 \x*~+l + a 3 + 1 ~ x + 1/ 

Integrating by Formulas (14) and (30), and reducing, w* 
ha^, 



METHODS OF INTEGRATION. l%\ 

Fourth Case. — When some of the binomial factors are 
imaginary and equal. 

In this case, the denominator will contain one or more 

factors of the form [(x — a) 2 4- b 2 ] n ; in determining th** 
partial fractions, we combine the methods used in the 
second and third cases. 

Let us assume, 

, _ x 3 dx 

y ~ (x 2 -2x + 2)"* 

Assume the identical equation, 

x 3 Ax + B Cx + D 



(1) 



(x 2 -2x+2) 2 (x 2 -2x+2) 2 x 2 -2x-\-2 

Clearing of fractions, and proceeding as before, we find, 

A =2, B = - 4, = 1, and D = 2. 

Substituting these in (1), multiplying by dx, and separating 
the fractions, we have, 

, _ 2xclx 4:dx xdx 

y ~ 7^2 O^. i 0\2 7^2 0~ i 0\2 "■* 



(x 2 - 2a; 4- 2) 2 (a 2 - 2a; + 2) 2 a; 2 - 2a; 4- 2 
2<fo 



a; 2 — 2x + 2 



(2) 



Making (x — l) 2 4- 1 = z 2 4 1. whence z* — 1 = 2, and 
r/a* = c?2, we have. 



_ 2(z + l)(/z _ ±dz (z 4- l)<fe 2dz 

V ~ (z 2 4 I) 2 ~ (z 2 4- l) 2 + z 2 4- 1 + z* + 1' 



or, 






>» 4- l) 2 (z 2 +l) f * f + 1 * § 4- 1 



128 INTEGRAL CALCULUS. 

The first, third, and fourth terms can be integrated by 

known formulas ; the second is a particular case of the 

dz 
form , which can be integrated by the aid of 

(z 2 -r l) n 
Formula D, yet to be deduced. 

Integrating the first, third, and fourth, and indicating 
the integration of the second, we have, 



V = - (^TI) - 2 fwTlY + I ^ + *> + 3tan"l, 



From what precedes we infer that all rational differen- 
tials are integrable; consequently, all differentials that 
can be made rational in terms of a new variable are also 
integrable. 

Integration by Substitution, and Rationalization. 

67. An irrational differential may sometimes be made 
rational, by substituting for the variable some function of 
an auxiliary variable ; when this can be done, the integra- 
tion may be effected by the methods of Articles 65 and 66. 
When the differential cannot be rationalized in terms of 
an auxiliary variable, it may sometimes be reduced to one 
of the elementary forms, and then integrated. The method 
of proceeding is best illustrated by examples. 



When the only Irrational Parts are Monomial. 

68. When the only irrational parts are monomial, a dif- 
ferential can be made rational by substituting for the 
primitive variable a new variable raised to a power whose 



METHODS OF INTEGRATION. 129 

exponent is the least common multiple of all the indices 
in the expression. 

Let . dy = (a; ' ~ rg3) — (1) 

1 -j-X 3 

The least common multiple of the indices is 6 ; making 

x = z 6 , we have, x* = z 3 , x 3 = 2 4 , x 3 = z 2 , and dx = 6z 6 de : 
these substituted in (1), give, 

dy = ±- — 6z 5 dz = — 6 — - — dz. 

y 1 + z 2 z 2 + 1 

Performing the division indicated, we have, 

dy =z — e(zt — z* — z 5 + z* + z 3 — z 2 — z + 1 



+ JTTT-- 



h) 



Integrating by known methods, and replacing z by its 
value, a?*, we have, finally, 

34 67 6432 

y = — jx 3 + - x® + x — -z* — -x* 

+ 2xi + dxi - 6xi - 31(1 + a£) 4- 6tan-^i 

When the only irrational parts are fractional powers 
of a binomial of the first degree, the differential can be 
rationalized by the same rule, and consequently can be 
in legrated. 

Let us take the expression, 

dy = (x + Vx + 2 -f tyx + 2) da ( 2 ) 

6* 



130 INTEGRAL CALCULUS. 

Assume, 

x + 2 = z 9 , whence, x = z* — 2, and dx = 6z & dz. 

Substituting in (2), we have, 

dy = (z* - 2 + 2 3 + z 8 )6z 6 ^ = G^ 11 - 2z& + z* + z 1 )^. 

Integrating, and substituting for 2 its value, (x + 2) *", wt 
have, 

y = \ (x + 2)i - %(x + 2) + | (x + 2)* + J (z + 2)*. 

Binomial Differentials. 

69. Every binomial differential can be reduced to the 

form, 

r 

dy = A(a + fo^V 7 * -1 ^; (1) 

in which w, w, r, and 5 are whole numbers, and n positive. 

Thus, the expression, (a? ■ + 2x~*)~ s x*dz, can be re- 
iuced to the form (1 4- 2x 4: )~^x^dx by simply removing 

the factor x~? from under the radical sign. If, in this 
result, we make x = 2 4 , whence, dx = ±z 3 dz, it will be- 
come, 4(1 + 2z)~*z 5 dz, which is of the required form. In 
like manner, any similar expression may be reduced to 
that foim. The constant factor, A, may be omitted during 

v 

integration, and the exponent of the parenthesis, -, may 

be represent ^d by a single letter p, as is done in Article 70. 
After integration, the factor may be replaced, and the value 

v 

of - may be substituted for p. 



METHODS OF INTEGRATION. 133 



FIRST CRITERION. 

Form (1) may be integrated by some of the methods 

v 

explained in Articles 65 and 66, when - is a whole number 

s 



SECOND CRITERION. 
171 

Form (1) can be integrated when — is a whole number. 

1 
For let us assume (a + bx n ) = z s , or, z = (a 4- bx n ) g \ we 
have, by solution and differentiation, 

„. i s m 

and, 

--1 

Substituting in (1), we find, 

r -—I 

(a + bx n )~»x m - l dx = ^(^— ) " f r+ *- 1 * •■■ (5!) 

Which expression is integrable, as was shown above, when 
— is a whole number. 



THIRD CRITERION. 

in t 
Form (1) can be integrated when 1- -, is a whole 

71 S 

number. 

For let us assume, 

, „ s » (a 4- bx n \% 
a + bx = z x , or, z = ( 1 . 



132 INTEGRAL CALCULUS. 

Solving and differentiating, we have, 
1 1 1 



x = (—^—Y = a n (z s -b) n ; x m =a\z 8 -b) n 
V - l> 

and, 

m m 

x m ~ X dx = - - a^(f- J)."*"" z*~ l dz. 



Substituting in (1), and reducing, we find, 

r m r m r 

{a -f bx n yx m ~ 1 dx = - - an + ~ s (zs - b)~^~ s ~ z r + s ~ l di 

(3) 

Which can be integrated, as was shown above, when 

\Yl T 

— V -, is a whole number. 
n s 



EXAMPLES. 

1. dy = (1 + x*Yx*dx (4) 

771 4 

Here, — = - = 2 ; hence, the second criterion is satis- 
fy Z 

fied. 

Comparing (2) with (4), we find, 

a = 1, b = 1, n = 2, m — 4, r = 1, s — 2, and 2 = (l + 3- ? )^ 
Hence, 



2 6 Z 3 

6" ~ 3 



(1 + s»)* __ (1 +x*)% _ {3x* - 2) (1 + X*)* 
5 3 ~ " 15 



METHODS OF INTEGRATION. 133 

2. dy = — = dx(l + x*)~ V* (5) 

x*(l + x 2 )? 

try) n» SI 

Heie, \- - = — - — -= — 2; hence, the third crite- 

n s 2 2 ' 

rioji is satisfied. 

Comparing (3) and (5), we find, 

a =1, b = 1, n = 2, m = — 3, r = — 1, s = 2, an<] 



2 = 



Hence, 



p dx p. z s Sz- z* 



3x 3 



Integration by Successive Reduction. 

70. "When an integral can be made to depend upon a 
simpler integral of the same form, it is evident that by 
successive repetitions of the process, we may ultimately 
arrive at a form that can be integrated by one of the funda- 
mental formulas. This method of integration is called 
the method by successive reductio?i, and is effected by for- 
mulas of reduction, some of which it is now proposed to 
investigate. 

1°. If in the expression for a binomial differential, we 
omit the factor A, represent the exponent of the paren- 
thesis by jo, and factor the result, we have, 

(a V br^x m -\lx = x m ~ n (a + bx n )V~ l dx (1) 



134 INTEGRAL CALCULUS. 

Let u = x m ~ n , and dv = (a + bx n fx n ~ x dx. 

Differentiating the first, and integrating the second bj 
Formula (29), we have, 

du = (m — n)x dx : and v = ^— 7 l tt— ; 

v ' ' bn(p + 1) 

But, (a 4 bx n f + 1 =(a + bx n f (a 4- bx n ) ; 

_ a(a 4 bx n f + (g + bx n fx n 
bn(p + 1) w(jo 4- 1) 

Substituting in Formula (28), we have, 
• " v &m(^ 4- 1) 

n{p + \)J v ' 

Transposing the last term to the first member, and re 
during, we have, 

nip +\)J (a + " X ' X aX ~ bn(p + \) 

_ a(m-n) P ^ny^-n-X^ 
bn(p 4- l)t/ v 

Dividing by the coefficient in the first member, we have, 

( fl 4- toPfaT- 1 ** = ( \ ( \ \ 

v ; b(pn + m) 

_ a{m-ji)r j^^m-n-1^ [A] 

h(pn + m)J v 



p 4 \)J v 



METHODS OF INTEGRATION. 135 

Formula A enables us to reduce the exponent of the 
variable without the parenthesis, by the exponent of the 
variable within the parenthesis, at each application. It 
fails when pn + m = ; but in this case, the third crite- 
rion, (Art. 69), is satisfied, and the expression may be 
integrated by a previous method. 

2°. The expression (a + bx n ) p may be placed under the 
form, 

{a + -bx n )V- X (a + bx n ), 

or, a(a + bx n f~ X + b{a + bx n f~ l x n ; 

hence, we may write the equation. 



f(a + bx n fx 



™~ m - x ax 



= of (a + bx n f~ X x m - X dx + bj\a + bx n f- X x m + ,l - x dz 

(2) 

The last term of this equation can be reduced by 
Formula A. Replacing m by m + n, p by p — 1, and 
multiplying by b, that formula becomes, 



^- 1 x m + n ~ 1 dx 



if [a 4- bx n f- 

= (JLL*W£ - -^-f{a + bx«?-\™- X dx. 
pn + m pn +m*7 

Substituting in (2), and uniting similar terms, we have, 

fa J- bx n fx m - 1 dx 

= (a+bx n fx m _P^L__r {a + bx n )P -l x m-l dx [B] 
pn + m pn + m J 



136 INTEGRAL CALCULUS. 

Formula B enables us to reduce the exponent of the 
parenthesis by 1 at each application. It fails in the same 
case as Formula A. When m and p are negative, we 
endeavor to increase instead of diminishing them. For 
this purpose we reverse Formulas A and B. 

3°. Reversing Formula A, and reducing, we have, 

f{a + bx n fx m - n ~ x dx 

= („ + MY +1 f- _ HP* + *) r (a + lx n fxm - hlx , 
a{m — n) a(m — n) •/ ' 

Replacing m by — m + n 9 we have, 

•/ v ' am 

b(pn-m + n)P ^^—+—1^ [<7J 

am J 

4°. Reversing Formula B, and reducing, we have, 

f(a + lx n f- x x m ~ x dx 

= _ (, + fa")V» £L±*/( B + fc^-l* 

In this, substituting — jt? +1 for p, we have, 

J(a + bx) x ax- an(p _ 1) 

_ m + n-npP + ^ ) -,*lj^-l dx [D] 

an(p — 1) J 



METHODS OF INTEGRATION. 



The mode of applying Formulas A, B, C, and D, will 
oe illustrated by practical examples. 
1. Let it be required to integrate the expression, 

( r a _ x 2 )$x 2 dx. 
In Formula A, making 

a — r 2 , i — — 1, n = 2, p = \, and m = 3, 
tfe find, 

j(r 2 - x 2 )h 2 dx = - ( r2 -* 2 ) 2x + r ^-J{r 2 - x 2 )%dx 

(3) 

In Formula B, making 

a = r 2 , ~b = — 1, n — 2, p = £, and m = 1, 
we find, 

J*( r t _ x *)idx = ( r * - x ^ x + r -^f(r* -x 2 )~$dx 

W 



But f(r 2 - x 2 ) 2 dx = /- — — = sin -1 

J J Vr 2 - x 2 



Substituting this in (4), and that result in (3), we hav» 

finally, 



J{r 2 r x 2 ) 



K.^- (r*-x*)iz 



r 2U2 _ X 2\\ x r t _ lZ 



138 

2. Let 



INTEGKAL CALCULUS. 

dx 



x s (x 2 - a 2 ) 2 



= (- a 2 + x 2 ) V S dx. 



In Formula 0, making a = — a 2 , # = 1, n = 2, /? = -- J, 
and m = 2, we have, 

But by Formula (24), 



dx 1 _ia: 

— = - sec -. 
„% a a 



Substituting in (5), we have, for the value of y, 



f 



dx 



o/ fi ovi 2a 2 z 2 2« 3 «' 

a; 3 (a: 2 — a 2 ) 2 



3. Let A, = ^— )8 = (*» + l)- 2 ^. 

In Formula Z>, making x = z, a = 1, J = 1, w = 2 
p = 2, and m = 1, we have, 



nt, 

/r 



r/j 



y^ 2 + i)- x dz 



dz _i 
- = tan *». 

+ 2 2 






METHODS OF I^TEGEATION'. 139 

Hence, we have, for the value of y, 

/dz z 1 i—l 

TfTTp ~ 2{z^+ T) + 2 tan * 

This is the method of integration referred to under the 
Fourth Case of rational fractions. By a continued ap- 
plication of Formula D, any expression of the form, 

dz 
m 2~\ n > can ^ e re ^uced to an integrable form. 

yCl -r Z j 

The following additional examples can be reduced to 
integrable forms by the application of Formulas A, B, G, 
and D. 

. , x*dx 

Ans . y = _«V-*!)^ »?*(..-«•)*+ »£ sin-** 

4 o o 



ft. *fy = («* — Z 2 )^#. 



^ws. ?/ = K%(a 2 — ^ 2 ) 7 + — sin -. 



A K%2 9 6 

6. dy = (1 + x 2 )~*x*dx. Ans. y — — — — (1 + x 2 )*. 
_ , z 3 cfc 3 a: 8 + 2 

„ 7 # 4 <£c re 3 — Sx 3 . _i 

9. #*/ = — -t4. ^tt*. y = , — 77 Sill & 

* (1-z 2 ) 2 y 2VT=^^ ^ 



140 INTEGKAL CALCULUS. 

1A , dx 2x 2 + \ lA i 

10 - dy = ^{l-x'fi V = -~te~ {1 ~ X) 

5°. Let it be required to simplify the expression, 

x n dx 
V2ax — x 2 

Changing the form and removing the factor x from under 
the radical sign, we have, 

(2a-x)~K n ~hx; 
in formula A, making, 

a = 2a, i = — 1, n = 1, p = — -J, and m = w + £, 

and reducing, we have, 

b(pn + m) = — n, and a(m — n) = 2a(n — £) = a(2n — 1) ; 

hence, 






71—1 /- - 

x V2ax — x 2 



(2n - \)aP x»- l ax 

*> J V2ax -x 2 



Formula E enables us to reduce the exponent n by 1, 
at each application ; if n is entire and positive, by a con- 
tinued application of the formula we ultimately arrive at 

the form, / — , which is equal to versin -. 

J V2ax—x 2 a 



i. dy 



2. dy = 



METHODS OF INTEGRATION. 14] 

EXAMPLES. 

xdx 



V%ax — x 2 



j±ns. y = — v <*ux — jo- -\- a versin 

X 2 dx 



Ans. y = — a/2## — # 2 + « versin - 

d 



V%x — X 

A X 

a irs ti - — 

2 v r 2 



4?zs. y = — V%% — x 2 + o versin :& 



3. dy = — :. 

A/2ir - a; 2 

. 2a; 3 + 5x + 15 /x r 5 _i 

^4^5. y = V2x — x 2 4- -versin z. 

2 



Certain Trinomial Differentials. 

71. Every trinomial differential that can be reduced to 
the form 

I 
(a + bx ± x 2 fx m dx (1) 

can be made rational in terms of an auxiliary variable, 
and consequently can be integrated, when p and m are 
whole numbers. 

When p is even, the expression is already rational; 
when p is odd, say of the form 2n + 1, the expression 
becomes, 

(a + Ix ± x 2 ) n (a + bx± x 2 )^x m dx (2) 



in which the only irrational part is Va + ox zkx 2 . There 
may be two cases ; first, when x 2 is positive, and sewvMyy 
when x 2 is negative. 



142 INTEGRAL CALCULUS. 

1°. When x 2 is positive. 

Assume, 



Va + bx + x 2 = z — x ; .*. a + bx + x 2 = z* — 2z# + *' 
striking out the common term x 2 , and solving, we have, 



and 



_ z 2 - a _ 2(z 2 + bz + a) ^ 

*-27T^ ; •'* ^~ (2* + 6) 2 ^ 



A / — r-v — : — £ z 2 + fo + a , ox 

V a + bx + x 2 = — - — --7 — (3) 

2z + b v ' 



Substituting these in (2), the result will be rational. 



EXAMPLES. 

1. dy = t = (1 + x + x 2 ) *dx (4) 

VI + x + x 2 

Comparing (4) with (1), we find, a = 1, and b = 1. 
Hence, from (3), we have, 

V1 + x + a; _ ^ + i , ana ax - ^ + 1)f ate. 

Substituting in (4), and making 2 = a; + Vl + # + j 2 , 
we hare, 

2<tfz 

^ = 27TT ; 

,\ # = H%z + 1) = J(2z + 1 + 2 Vl + x + aT) 
« , dx 



V* 9 - z - l 



,4^, y = Z(2z - 1 + 2aA* - x - 1). 



8. dy = 



METHODS OF INTEGKATION. 143 

dx 



Ans. v = 



x Vl + X + X 2 

7/ 3X v 

\2 + x + 2 VTT^ + x 2 ) 
i°. When x 2 is negative. 

Let a and 3 be taken so as to satisfy the equation, 

a + bx - x 2 = (x — a)(0 - a) (5) 

Assume 



Va -\- bx — x 2 = V(% — a)(# — z) == (# — a )«; 

squaring and reducing, we have, 

/ h x 

(3 - x = (x - *)z 2 ; or, z - A/ ^zr~ a > 

hence, 

(3 + az 2 _ 2(tf - «)z<fe 






efa; = 



1 +z 2 ' (1 + z 2 ) 2 ' 

and 

V«"W-^ = ^g? (6) 

These values substituted in (1), will make it rational 



Let <fy — 



EXAMPLE. 

dx dx 



\/2- z -a; 2 V(z + 2)(1 - x) 

.•. a = — 2. and /3 = 1 (7) 



144 INTEGRAL CALCULUS. 



From (6), we have, 

, 6zdz j ,- 3s 

ax = rrr, and V2 — a; — x* = 



1 + z 2 ) 2 ' (!+**)' 

also 3 - 
Substituting in (7), we find, 




x + % 



dv = ~ T^fli 5 ••• 2/ = ~ ^tan x « = - 2tan ^i/ 1 —^ 



Integration by Series. 

72. It is often convenient to develop a differential into 
a series, arranged according to the ascending powers of 
the variable, and then to integrate each term separately. 
This method sometimes enables us to find an approxi- 
mate value for an integral, which cannot be found in any 
other manner ; it also enables us to deduce many useful 
formulas. 

EXAMPLES. 

i. dy = x*(l - xrfdx. 

Developing (1 — x 2 y by the binomial formula, and 
multiplying each term by x 2 dx, we have, 



qt4 /y6 -7*8 



•'' y ~ 3" ~ 10 ~ 56" ~ 144 ~ etC 



dp =t-^- s =dx(l +x*)~\ 
1 4- x % 



METHODS OF INTEGRATION. 145 

By Formula (14) y is equal to tan - x. Developing bj 
the binomial formula, we have, 

(1 + x 2 )~ X = 1 — x 2 + x± — x 6 4- x 9 — x 10 + etc. 

Substituting, and integrating, we have the Formula. 

_i x 3 x 5 x 1 x 9 
tan x =xc — — + — + g- - etc. 



3. dy = -r-^-- — dx(\ + a;) * 



= (1 — x + a; 2 — x 3 + etc.)e?a; 



/)•» /v>3 o^4 

.% y = Z(l 4- a;) = a; - ~ + — - j- + etc. 



4. dy = - Jt_ = dx{l _ ^ 2) -i 
Vl - z°~ 

. _i x 3 3x 5 3.5a: 1 



Integration of Transcendental Differentials. 

73. A transcendental differential is one that is expressed 
m terms of some transcendental quantity. There are three 
principal classes; viz., logarithmic, exponential, and cir- 
' mlar. 

Logarithmic Differentials. 

74. A large n imber of cases come under the general 
form, 

dy = x m -^(lx) n dx (1) 



146 INTEGRAL CALCULUS. 

Assume, 

u = (lx) n , and dv = x m dx; 

n{lx) n ~ l dx, , r r * 

:. da = — — - — and v— -■■ 

X )h 

substituting in Formula (28), and reducing, we have, 

y=fx m -\lx) n dx = *^M! _£y>-l(fc)— !*;..[>) 

Formula F is a formula of reduction ; it reduces the 
exponent of (Ix) by 1 at each application. 

EXAMPLES. 

1. dy = x(lx) 3 dx. 

Am. y = £(fc)t - *£(&,)• + ^(te) - *£. 

2. cfy = x 3 (lx)dx. Arts, y = — j— ^ — ^-. 

3. dy = x 3 (lx) 3 dx. 

Am. y = J((fe)» - |(fc)» + |(fr) - |) 

Reversing Formula F, and reducing, Ave have, 

fx m -\lxy- X dx^ X ^^ -™-fx m -\lx) n d-, 
Replacing n — 1 by — w, and reducing, we have, 

(Ixf (n - 1) (Ixf- 1 + n - 1 J '(tef- 1 

[ff» 



METHODS OF INTEGRATION 147 

The continued application of formula G gives as a final 

/x m ~ 1 dx 
— -j-^ — , which can be integrated 

by series. It fails when n — 1. Let m — 0, and n = 1 , 
we then have, 



Exponential Differentials. 

75. Let it be required to integrate an expression of the 
form, 

dy = x m a x dx (1) 

Assume, 

u = x m , and dv = a x dx ; .*. du = mx m dx, and y = — . 

Substituting in Formula (28), 

fx m a x dx = ^f- ~ a fx m ~ x a x dx \E\ 

When m < 1, we have, 

/a x dx _ a x la p x dx . . . 



148 INTEGKAL CALCULUS. 



EXAMPLES. 



1. dy = xa x dx. Ans. y = — f x — — J- 



2. dy — x a x dx. 

a x f s Sx 2 6x 6 \ 

S.dy= w . Ans. y=--+J — 

But by means of McLaurin's Formula, we find, 



x x* x° x* 

3 = 1+a;+ L2 + lX3 + L^4 + etC - ; 



hence, 



x dx _ ffdx 
x J \ x 



xdx \ 

+ dx + — — + etc. J 



lx + x + TM + TJ3l + etc -' 



which, being substituted above, gives, 






4. % = x 2 e x dx. Ans. y = e x (x 2 - 2x 4- 2). 



Circular Differentials. 

76. Circular differentials may be integrated by the 
methods of transformation and successive reduction, or 
they may be reduced to algebraic forms by making 
mix = z. 



METHODS OF INTEGRATION. 149 



EXAMPLES. 



1. Let dii = -7-^-. 
sins 



Prom trigonometry, we have, 

sinz = 2sin-Ja;cos£a: = 2tanJa'cos*|a;, 

dx _ dx d(t&\i\x) 

sina; 2tanJa:cos 2 Ja; tan^-a; ' 

Hence, 



/4| = ? (tani*) (1) 



2. Let dy = . 

cosa; 

If we make x equal to 90° — x in the preceding formula 
,ud reduce, we have, 

fj!L = - Z[tan(45° - &)] (2) 

J cosa; L v * /J 



3. Let dy = . 

tana; 



From trigonometry, we have, 

sin 2 dx dxcosx d(smx) 

tana; = ; or, = — : = — ^ -. 

cosa; tana; sina; sma; 

Hence, 

f£L= 1 ^ < 3 > 



150 INTEGRAL CALCULUS. 

4. Let dy = — — . 

a cote 

Making x equal to 90° — x in (3), and reducing, we ha?e, 

f£k = ~ 1 ^ <*> 

5. Let dy — - . 

smzcosz 

Prom trigonometry, we have, 

sinzcosz = Jsin2a ; 

we also have, 

dx = %d(%x) ; 

hence, 

dx _ d(2x) 
sinzcosjc sin2# 

Applying Formula (1), we have, 

f dx = l{ttmx) ... . (5) 
•7 sinzcosz ' 

Let us have the expression, 

dy = $m m x(>o$ ll xdx (6) 

Making sinz = z, whence cosz — Vl — z 2 , 

and dx 



Vl-z* 
we have, 

71-1 

dy --= z m (l -z*) 2 dz (7) 

Form (7) can always be integrated when m and n are 
whole numbers, because it will then satisfy one of the three 
criterions (Art. 69). 



METHODS OF INTEGRATION. 151 

6. dy = sm 2 £cos 3 xY££. 

Comparing with (6), w* 1 find ?>? = 2, and n = 3; hence, 
from (7), we have, 

z' 6 z s 
dy = (1 — z 2 )z 2 cfc; /. y = — = sin 3 z(-|- — |sin 2 z). 

•"» o 

7. Let dy = sm 3 xdx. 

Here m = 3; and n = 0; and d# = (1 — z 2 )~ *z 3 dz ; 
making x = z, a = 1, J = — 1, n = 2, p = — J, and ?w = 4, 
in Formula ^4, we have, 

yii - s 2 )- Ws = - (i ~g 2) ^ 2 + iy*(i - »•)"*«& 

But from Formula (29), 

J (I - z 2 )-hdz= - (1 -z 2 )*; 
hence, 

y — -* - (1 - z 2 ) V — -(1 - z 2 )i — _ -cosz(sin 8 :z + l\ 

o o o 



8. Let rfy = cos 2 2*fo = dzVl — z 2 . 
Applying Formula (B), we find, 

Bin, 



fjT^)i=fi* = 



X, 



152 INTEGRAL CALCULUS. 

Hence, 

= 2' 



y = -(coszsmx + x). 



9. Let dy = -—- - = - — = (1 - z 2 ) V 8 <fe 



Applying Formula C, we have, 

/a—) - - 

4- 

2 



* 2 -, (fe = _(l^£.!)k! 



i--J\l-z*) *«-** 



But, 



/(l - 8») *S 


na-/ 1 A 


/Vfa 


= Z(tan£z). 


J^ ' 


t/Vi - 


Z 2 Jsmx 


Hence, 


cosz 
y " 2sin 2 z + 


-?(tanjz). 

ii 






10. Let % = 


dx 


dz -a 


-*■)" 


"V-** 


siniccos 3 :/: ~~ z(\ 


-z 2 r~- {1 


By Formula D, 


we have, 









' /(l - z«) 'V»& = i (1 -2 2 ) * + T(l - z 2 ) V^dfe. 



But, 



Vl — 2 2 Vl - 2 S 



= /- = ?(tan#) 

^7 sinacosa; 

Hence, 



^ = aii +z(tana:) ' 



METHODS OF INTEGRATION. 153 

11. Let dy = tan^xclx = (1 - z*)~**z*dz, 

y = Jtan 3 # — tana; -f x. 

EXAMPLES IN" SUCCESSIVE INTEGRATION. 

12. d 2 y — sina?cos 3 3%£c 2 = smx(ch'mx) 2 . 
Making sinz = z, whence, d 2 y = z(dz) 2 , we have, 

dt/ <3 2 z^ 

^ = g- + G > and y = g- + Cz + ^ 



or, 

sin 3 re 



4- Osinz + C 



* 6 

13. d 2 y = coszsin 2 :^ 2 = cos£(cfcosaj) 2 ; 

and 



rt COS*£ + C\ 

d(cosx) 2 



v = -z cos 3 z + Ceoscc 4- C. 
b 



14. d 2 ?/ = - — ; 



T^ = - + V ; or,y= Ix + 6fe+f? 
dx x ' 3 



Integration of Differential Functions of two Variables. 

77. A total differential of a function of two variables is 
of the form, 

dz — Pdx + Qdy (1) 

7* 



154 IKTEGitAL CALCULUS. 

In which P and Q are functions of x and y ; but differ- 
ential expressions of that form are not necessarily total 
differentials. That they may be so, they must (Eq. 5, 
Art. 26), satisfy the condition, 

d_P = dQ _ 

dy dx * ' 

When an expression of the form (1) is to be integrated, 
we first apply the test expressed by (2) ; if that be satisfied, 
the expression is integrable. To perform the integration, 
integrate one of the partial differentials with reference to 
its corresponding variable, that is, as though the other 
variable were constant, and add such a function of that va- 
riable as will satisfy equation (1). Thus, to integrate ex- 
pression (1), we have, 



■/ 



Pdx + R (3) 



n which R is a function of y alone. The value of R may 
be found by differentiating the second member of (3) with 
respect to y, and placing its partial differential coefficient 
equal to Q. Hence, we have, 



rf/ 



p dx dR 
4- 



dy dy 

Transposing, multiplying by dy, and integrating with 
respect to y, that is, as though x were constant, we find, 



AdfpdxX 



dy 



)dy- 



METHODS OF INTEGRATION. 155 

Substituting in (3), we find, 

/P ( dfPdx\ 
Pdx+J {Q-J lj —)ay (4) 



EXAMPLES. 

1. dz = 3x 2 y 2 dx + 2x 3 ydy. 

Here P = 3x 2 y 2 , and Q = 2x 3 y, and (2) is satisfied 
Integrating by Formula (4), we have, 

z = x 3 y 2 4- / {2x 3 y — 2x 3 y)dy = x 3 y 2 -+- C. 

2. dz = ydx + xdy. 

The test (2) is satisfied. Integrating by (4), we have, 

z = yx + I (xdy — xdy)dy — yx-\- (\ 

By Formula (4), we have, 

V J V J y 2 y 2 J J >i 
4. dz — (6xy — y 2 )dx + (3a; 2 — 2xy)dy. 

5 = &£* y — y 2 x + f(3x 2 — 2xy — 3x 2 + 2z#)ffy 

= 3x 2 <j - y*x + 



PART IV. 

APPLICATIONS OF THE INTEGRAL CALCULUS 



I. Lengths of Plane Curves. 

Rectification. 

78. Rectification is the operation of finding an expres- 
sion for the length of a curve. This may be done by 
finding an element of the length, as explained in Art. 51, 
and then integrating it between proper limits. If we 
apply the integral sign to Equation (1), Art. 51, we have, 

L =JVdx* + % 2 (1) 

l°. To rectify the semi-cubic parabola, wliose equation is, 
y z = a 2 x 2 . 
Differentiating and substituting in (1), we have, 

L = \- a f{^' + ^dy (2) 

Integrating by Formula (29), we have, 

L = ±-(i«* + 9y)>+ G (3) 

4 t (I 

As explained in Art. 60, this integral is indefinite, and 



LENGTHS OF PLANE CURVES. 157 

expresses the length of an arc from any ordinate up to any 
other ordinate. 

If we estimate the length from a point whose ordinate 
is 0, the length at that point is 0, and, from (3), making 
L, and y = 0, we have, 

Substituting this value of O in (3), and denoting the cor- 
responding value of L by L\ we have, 

L ' = k^ + ^-w (4) 

This is a particular integral, and it expresses the length 
of the arc from the particular ordinate 0, up to any 
ordinate y. 

If we wish to find the length from the particular ordi- 
nate 0, up to an ordinate b, we make y = b, in (4). Doing 
so, and denoting the corresponding value of L' by Ii\ 
we have, 

L " = k^ + u ^-w ^ 

This is the definite integral, and expresses the length of 
the arc from the ordinate 0, up to the ordinate b. 

In this case the limits are and b ; the integral may be 
found otherwise, as follows: 

Making y = 0, in (3), and denote the corresponding 
value oi L by Lq 9 we have, 

L o = k {ia * )i+0 (6) 



L58 INTEGRAL CALCULUS. 

Making y — b, in (3), and denoting the corresponding 
value of L by L b , we have, 

Equation (6) expresses the length of the curve from any 
point up to the point whose ordinate is 0, and (7) ex- 
presses the length of the arc from the same point up to a 
point whose ordinate is b ; the first taken from the second 
will therefore be the definite integral, and will express, as 
before, the length of the arc from the point whose ordinate 
is 0, up to the point whose ordinate is b. Making the 
subtraction, and adopting the notation explained in Art. 60, 
we have, 

b 

L " = i/ (4aS + 9 * = k ^ + 9i >*- w • ■ (8) 

o 
The same result as found in (5). In this case the initial 

abscissa is and the terminal abscissa is - . 

a 

2°. To rectify the cycloid, whose differential equation 

(Art. 56), is 

dx=± -J^ = (9) 

V%ry — y 2 

Substituting, in (1), and reducing, we have, 

L = V2?J{2r -y)~hy (10) 

Integrating between the limits and 2r Formula (29), 

wo have, 

2r 

rr = V^/V -y)~*dy = te (ID 



LENGTHS OF PLANE CURVES. 159 

This is the expression for half of one branch. Hence, the 
whole branch is equal to eight times the radius of the 
generating circle. 

3°. To rectify the circle, whose equation is, 

y — (r 2 — x 2 )^. 

Differentiating, substituting in (1), and developing bj 
the binomial formula, we have, 

L = rl (r 2 — x 2 ) ^dx =1 (clx + — — . x 2 dx + " . . x*dx 
J v ' J 2r 2 %Ar*> 

Performing the integration, making r = 1, and com- 
mencing the arc at the point whose abscissa is 0, that is, 
at the upper extremity of the vertical diameter, we have, 

v = x + 3 V + Si xi + mrf* + etc (13) 

Formula (13) may be used for finding the length of an 
arc, but the series is not very converging, and therefore 
the process is tedious. We know, when x = \, that L' is 
equal to an arc of 30°, that is, to £ir. Making L' = £*-, 
and x — \, we have, 

111 L3 1.3.5 KOOKftDft 

6* = 2 + 2KM + WH + 2^4Z7 + ^ = * 5235987 ; 

.-. * = 3.1415 (14) 

4°. A better method for deducing the value of *, is to 
find the length. of the arc, in terms of its tangent. 

If L = tan - x, we have (Art. 18), 

dL = TT* = V- + x * rW 



i60 INTEGKAL CALCULUS. 

Developing by the binomial formula, we have, 

dL = dx{l — x2+x*-x*+x*- etc.) (15) 

Integrating, and determining G, as before, we have, 

f3 /~f5 tyl sy>9 

To render (16) converging, x must be made small 
Assume the trigonometrical formula, 

. tana + tan£ 

tan (a 4- b) — -. -. — - v 

1 — taiu/tun/; 



u — tan m, u — iitu /t, ctnu u, ~r u — \Jixu 
then will 



and let a = tan 1 m, b = tan n, and a + b = tan s, 



^ = 1 ; or, % = ■ — - (17) 

1 — mn mz +1 ' 

1 2 

rf 2 = 1, and m = -, we have, ?£ = - ; 
5 d 

.*. tan -1 1 = tan -1 - + tan - -. 
5 o 



2 1 7 

-, and m = -, we have, n = — 
o o 1 7 



, _i2 , _il , _i 7 
'. tan - = tan - + tan — . 
6 o 17 



7 1 9 

If g = — - and m = -, we have, w = — - 
17 5' 46 



tan -t. — tan - + tan — „ 
17 5 4b 



ti> anC ' w = 5' We haV6 ' " = ~ 2^9 



-1 9 _i 1 _i 1 

tan - = tan --tan m 



AREAS OF PLANE CURVES. If 1 

Hence, by successive substitution, we find, 

7 = tan -1 1 = 4tan _1 - - tan -1 -i- (18) 

If we make x = -, in Equation (16), we find the value ol 
o 

San r, and in like manner, we find the tan — r; sub- 

O Zoo 

stituting them in (18), and reducing, we have the value of 
r. With very little labor, the value of <k may be found to 
8 or 10 places of decimals. 



II. Areas of Plane Curves. 

Quadrature. 

79. Quadrature is the operation of finding an expression 
for the area of a portion of a plane bounded by a curve, 
the axis of abscissas, and any two ordinates. This is done 
by finding an expression for the elementary area, as ex- 
plained in Art. 51, and then integrating the result between 
proper limits. Applying the sign of integration to For- 
mula (2), Art. 51, we have, 



= I ydx 



(i) 



1". To find the area of a parabola, whose equation is, 
y 2 = 2px. 
Finding the value of y, and substituting in (1), we have, 

A = \/2p fx^dx = ?\/2p . as* + (2) 



Lt>2 INTEGRAL CALCULUS. 

If we commence the area at the ordinate 0, we have 
O = 0; hence, 

A' = \^.x§ = lyx . (3) 

That is, the area of any portion of a parabola, reckoned 
froin the vertex, id two-thirds the rectangle of its terminal 
Co-ordinates. 

2°. To find the area of a circle, whose equation is. 



y = Vr- X 9 . 
tf instituting in (1), we find, 



A = fir 2 - x 2 )Klx (4) 



deducing by Formula B, and integrating the last term 
by Formula (20), we have, 



= -i '- + — sin - + G 



(5) 



Taking the integral uecween the limits, x = and x = r, 
which gives the area of a quadrant, and remembering that 

sin"" 1 — sin - = J-t, we have, 

A'' = ^-rsin -1 1 - sin -1 0] = ^-; .-. Area = cr* . . . (6) 
2 l J 4 

3°. To find the area of the ellipse, whose equation is, 
y — -(a 2 -j 2 ) 2 . 



AKEAS OF PLAN"E CURVES. 163 

Substituting in (1), and integrating, as in the last example, 
we have, 

Integrating from x — to x = a, and multiplying by 4, we 
Lave, for the area of the ellipse, 

1A" =*ab (8) 

4°. To find the area of the cycloid, whose differential 
equation is, 

V%ry — y 2 
Substituting in (1), we find, 

y 2 dy 



:f- 7 m= (9) 

J V%ry — y 2 



\2ry — y 2 

Applying Formula E twice, and Formula (26) once, we 
find, 



yV2ry - y 2 
A- 

+ o"[~ — V%ry — y 2 + rversin -1 -] + C. 

Taking the integral between the limits y = 0, and y = 2r, 
and multiplying by 2, we find, for the area of the cycloid. 

2A" = d«r 2 (10) 

That is, the area between one branch and the directing 
line is three times the area of the generating circle. 

5°. To find the area of the logarithmic curve, whom 
equation is, y = Ix. 



164 INTEGRAL CALCULUS. 

Substituting in (1), we have, 

A =J(lx)dx .... (11) 

Making m = 1 and n = 1, in Formula F, and reducing, 
we have, 

^. = zfc — x + (7 (12) 

IlVwe commence the arc at the ordinate corresponding to 
r. — l, we have, (7=1, and 

-4' = s(fe - 1) + 1 (13) 

6°. To find the area of a rectangular hyperbola, bounded 
by the curve, one asymptote, and any two ordinates to that 
asymptote. 

Assume the equation, 

xy = m, and make m = 1, whence y = -. 

x 

Substituting in (1), we have, 

A =f—= te+C (14) 

Commencing the area from the ordinate through the vertex, 
where x — 1, we have, (7=0; hence, 

A' = Ix (15) 

That is, the area commencing from the ordinate through 
the vertex is the Napierian logarithm of the terminal ab- 
scissa. Had we not made m = 1, we should, m like man- 
ner, have found 

A' — mix. 

In which the area is equal to the logarithm in a system 
whose modulus is m. 



AREAS OF SURFACES OF REVOLUTION. L65 

III. Areas of Surfaces of Revolution". 

Surfaces Generated by the Revolution of Plane Curves. 

80. The area of a portion of a surface of revolution, 
bounded by two planes perpendicular to its axis, is de- 
termined by finding an expression for an elementary zone, 
as explained in Art. 51, and integrating the result between 
proper limits. Applying the sign of integration to For- 
mula (3), Art. 51, we have, 

S=j%«yVdx* + dy* (1) 

1°. To find the surface of a sphere, the equation of the 
generating circle being, y = (r 2 — x 2 y. 

Differentiating, we have, dy = — (r 2 -- x 2 )~\xdx ; sub- 
stituting the values of y and dy in (1), and integrating 
from — r to -f r, we have, 



rfdx = ±*r 2 (2) 



2<7T7 



Hence, the area of the surface of a sphere is equal to four 
great circles, or to two- thirds the surface of the circum- 
scribed cylinder. 

2°. To find the surface of a right cone. 

The equation of the generating line, the vertex of the 
cone being at the origin, is 

y = ax ; .-. dy — adx. 



'66 IKTEGEAL CALCULUS. 

Substituting in (1), and reducing, we have, 

S = 2*aVl + a 2 Jxdx = vax 2 Vl + a 2 + G (3) 

If the initial plane pass through the vertex, we have, foi 
that plane, S = 0, and x = 0, whence (7=0; hence, 



S' = vax 2 Vl + a 2 = «y X xVl + a 2 (4) 

But a is the tangent of the semi-angle of the cone, and 
consequently xVl + a 2 is the slant height; 2iry is the 
circumference of the cone's base; hence, the convex sur- 
face is equal to half the circumference of the base into the 
slant height. 

3°. To find the surface of the paraboloid of revolution. 
Assume the equation y 2 = 2px, whence, 

y = V2px, and dy 2 = £- dx 2 . 
Zx 

Substituting in (1), and reducing, we have, 

Cj = 2* fip* + 2px)idx = ^{p 2 + 2px)i + G (5) 

[f the initial plane pass through the vertex, we have, 

Which, in (5), gives, 

S' = j[±(p* + 2 P x)§-p*] (6) 

4 e . To find the surface generated by revolving one branch 
of a cycloid about its base. 



VOLUMES OF SOLIDS OF REVOLUTION. 167 

Assuming tlie equation, dx = =h ^ — ^ ? and substi 

V'&ry—y* 

tuting in (1), we have, 

N = 2irA/2r/(2r - y)~^ydy (7) 

Applying Formula A, and integrating the last term by 
Formula (29), we have, 



6' = - 2* Va7[J y(2r - y)* + y (8r - 2/)*] + 6'. 

Integrating between the limits y = and y = 2r, we have, 



S"= 6 -£«r* (8) 

This is the surface generated by half of one branch. 
Hence, to find the whole surface, we multiply by 2. This 
gives 



IV. Volumes of Solids of Revolution. 

Cubature. 

81. Cubature is the operation of finding the volume of a 
solid. When this is bounded by a surface of revolution 
and two planes perpendicular to its axis, the volume may 
be ascertained by finding the volume of an element 
bounded by two such planes infinitely near to each 
other, as explained in Art. 51, and then integrating thf 



168 INTEGRAL CALCULUS. 

result between proper limits. Applying the sign of inte- 
gration to Formula (4), Art. 51, we have, 



V = «j y 2 dx (1) 



1 w . To find the volume of a sphere. 

Assume the equation, y 2 = r 2 — x 2 , and combine it with 
1) ; we have, 



=/(,- 



dx — x 2 dx) = tt r 2 x — — + (7. 



Taking the integral between the limits, x = — r, and 
x = -\- r, we find, 

r 2r 3 ~\ 4 1 
Y" = «\2r* - ^-J = ^ 3 = 4ff » X gr (2) 

That is, tf/ae volume is equal to the surface by one-third the 
radius. 

2°. To find the volume of a spheroid of revolution. 

There are two species of spheroids of revolution. 

1st. The prolate spheroid, generated by revolving an 
ellipse about its transverse axis. 

2dly. The oblate spheroid, generated by revolving an 
ellipse about its conjugate axis. 

1st. The prolate spheroid. In this case the equation of 

I 2 
the meridian curve is, y 2 = —{a 2 — x 2 ). Substituting 

a 

this in (1), and integrating between the limits x = — a, 

and x — + a, we have, 

+ a 

V" = 7T b -^-f(a 2 - x 2 )dx = %rb 2 a = %rb 2 X 2a (3) 

—a 



VOLUMES OF SOLIDS OF REVOLUTION. 169 

That is, the volume is equal to two-thirds of the circum- 
scribing cylinder. 

2dly. The oblate spheroid. In this case, if the conjugate 

axis coincide with the axis of x, the equation of the meri- 

a 2 
dian curve is, y 2 = — (b 2 — x 2 ). Substituting in (1), and 

integrating from — b to + b, we have, 

V" = %ra 2 b = \«a 2 X 2b ..... (4) 

Hence, as before, we have, the volume equal to two-thirds 
the circumscribing cylinder. 

In both cases, if a — b = r, we have, 

V" = %«r* . . . . (5) 
o 

This hypothesis causes the ellipsoids to merge into tne 
sphere. 

3°. To find the volume of a paraboloid of revolution. 

The equation of the meridian curve is, y 2 = 2px. Hence, 
from (1), we have, 

V = 2*plxdx = irpx 2 + C (6) 

If the initial plane pass through the vertex, we have 
C = 0, and 

V = «px 2 = *y 2 X \x . . . . (7) 

That is, the volume is equal to half the cylinder that has the 
same base and the same altitude. 



170 INTEGRAL CALCULUS. 

4°. To find the volume generated by revolving one branch 
of the cycloid about its base. 

The differential equation of the meridian curve is, 
dx- ydil 



V%ry — y* 
hence, from (1), we have, 

y*dy 



J V%ry — y (> 



Keducing by Formula F, integrating the last term by For- 
mula (26), and taking the integral between the limits 
y — 0, and y = 2r, we find, for one-half the volume 
required, 

V" =l« 2 r s , or2F" = 5* 2 r3 =!*(2r)' X 2*r (8) 

4i o 

Hence, the volume is equal to five-eighths the circumscribing 
cylinder. 

5°. To find the volume generated by revolving the lugarith 
mic curve about the axis of numbers. 

The equation of the generatrix is y = Ix. Hence, 
V=*f{lx)*dx (9) 

Reducing by Formula F, we have, 

V = «[x{lx)* - 2(xlx - x)] + G. 

If the initial plane pass through the point whose 
abscissa is 1, we have, C = — 2*. Hence, 

V = «[x(lx)* - 2(xlx -2 f 1)] (10) 



PART V. 

APPLICATIONS OF THE DIFFERENTIAL AND 
INTEGRAL CALCULUS TO MECHANICS AND 
ASTRONOMY. 



I. Centre of Gravity. 

82. In what follows, bodies are supposed to be homoge- 
neous ; the weight of any part of a body is therefore pro- 
portional to its volume, and consequently the weight of 
the unit of volume may be taken as the unit of weight. 
Points, lines, and surfaces are supposed to be material : 
A material point is a body whose length, breadth, and 
thickness are infinitesimal ; a material line is a line 
whose length is finite, and whose breadth and thickness 
are infinitesimal; a material surface is a body whose 
length and breadth are finite, and whose thickness is infi- 
nitesimal; under this supposition a point is an elementary 
portion of a line, a line is an elementary portion of a sur- 
face, and a surface is an elementary portion of a solid. 

The weights of the elements of a body are directed 
toward the centre of the earth, and because the bodies 
treated of are exceedingly small in comparison with the 
earth, these weights may be regarded as a system of par- 
allel forces ; hence, the weight of a body is equal to the 
sum of the weights of its elements and is parallel tc 
them. 



172 DIFFERENTIAL AND INTEGRAL CALCULtJ8. 

The centre of gravity of a body is a point through which 
its weight always passes. This point may be found by the 
principle of moments, which may be enunciated as follows : 
the moment of the resultant of any number of forces, with 
respect to an axis, is equal to the algebraic sum of the 
moments of the forces, with respect to the same axis. 
(Mechanics, Art. 35.) 

In applying this principle, we assume the following re- 
sults of demonstrations in mechanics : 1°. The centre of 
gravity of a straight line is at its middle point; 2°. The 
centre of gravity of a plane figure is in that plane ; 3°. If 
a plane figure have a line of symmetry, its centre of gravity 
is on that line ; and 4°. If a solid have a plane of symmetry, 
its centre of gravity is in that plane. (Mechanics, Arts. 44, 
45, 46.) 

To deduce general formulas for finding the centre of 
gravity of a body, assume a system of co-ordinate axes that 
are to retain a fixed position with respect to the body, but 
that change position when the body moves; denote the 
volume, and consequently the weight, of any element of 
the body by dv, and the co-ordinates of its centre of 
gravity by x, y, and z; denote the weight of the body by 

v = I dv, and the co-ordinates of its centre of gravity by 

x„y 19 and s r 

If the body be placed in such a position that the plane 
xy is horizontal, the weights of the elements and of the 
body are parallel to the axis of z, the moment of the body, 

with respect to the axis of y, is x t l dv, the moment 

of any element, with respect to the same axis, is xdv. 



(1) 



Vi 



APPLICATIONS TO MECHANICS AND ASTKONOMY. 173 

and the algebraic sum of the moments of all the elements 
is I xdv; hence, from. the principle of moments, we have, 

/p fxdv 

lv — Ixdv : .\ x« — —x — . . , 
J jdv 

In like manner, we have, 

/» p fydv 

jdv =Jydv ; .-. y t = J j- (2) 

If the body be turned about so that the plane yz is 
horizontal, we have, in like manner, 

/» p Izdv 
dv = Izdv ; .'. z t = -^ — (3) 
*/ jdv 

When the body is in a plane, that plane may be taken 
as the plane xy, in which case z t = ; if the body have an 
axis of symmetry, that may be taken as the axis of x, in 
which case z t = 0, and y t = 0. 

Centre of Gravity of a Circular Arc. 

83. Let the radius perpendicular to the chord of the arc 
be taken as the axis of x ; then will z t , and y^ be equal tc 
0. Denote the radius of the circle by r, ^ A 
the chord by c, and the are by A. The 
origin being at the centre, the equation of 
the arc is, y 2 = r 2 — x 2 ; hence, 



y& 



dv = Vdx 2 + dy 2 — \/ ±-dy 2 + dy 2 

_ rdy _ rdy 



.n ., 



c 

Fig. W 



X ^/ r 2 _ yt 

12* 



174 



DIFFERENTIAL AND INTEGRAL CALCULUS. 



Substituting in (1), and integrating between the limits, 
y =: — j£, and y = + Jc, we "have, for the numerator, 



/ rJy = re, 

and for the denominator, 
*f r*M_ = J^-l £_„•„-: 



Hence, 



^ 



re 



arc4£C 



, or arc .4 #6' : o : : r : x v 



That is, /Ae centre of gravity of the arc of a circle is on the 
diameter that bisects the chord, and its distance from the 
centre is a fourth proportional to the arc % its chord, and 
the radius. 



Centre of Gravity of a Parabolic Area. 

84. Let the area be limited by a double ordinate, and 
denote the extreme abscissa by a. From the equation of 
the curve, y 2 = 2px, we have, 

y = <y/%> • x*. 
,\ dv = ydx = V2p . x%dx, 

and xdv = V%p • x%dx. 
Substituting in (1). and integrating between 
the limits and a, we have, Fig. n. 




*, =5* 



APPLICATIONS TO MECHANICS AND ASTRONOMY. 175 

That is, the centre of gravity is on the axis, at a distance 
from the vertex equal to three-fifths the altitude of the 
segment. 

Centre of Gravity of a Semi-ellipsoid of Revolution. 

85. Let the axis of the ellipsoid be taken as the axis 
of x. Then, if the origin be taken at the centre, the 
equation of the generating curve is 



b 2 



In this case, we have, 



x 2 ). 




dv = <ry 2 dx 



■rr—(a 2 -x 2 )dx, 
a z 



Fig. 18. 



and xdv = <* — {a 2 x — x 3 )dx. 



Substituting in (1), and integrating between the limits 
x = 0, and x = a, we have, 



That is, the centre of gravity of a semi-prolate spheroid 
of revolution is on its axis of revolution, and at a distance 
from the centre equal to three-sixteenths the major axis of 
the generating ellipse. 

If we change a to b, and b to a, we find for the sour 
ablate spheroid, 

3 



<=\b 



1A 



2b. 



l7ti DIFFERENTIAL ANT) INTEGRAL CALCULUS. 

Centre of Gravity of a Cone. 

86. Let the axis of the cone be the axis of x, and tht 
vertex of the cone at the origin ; denote the altitude by h, 
and the radius of the base by r ; then will the tangent of the 

T 

semi-angle of the cone be j, the equation of the generating 

T 

line will be y = j x, and we have, 

dv = iry z dx = ir — x 2 dx, and xdv = « — x 3 dx. 

Substituting in (1), and integrating between the limits 
and h, we have, 

x 1=l h. 

Centre of Gravity of a Paraboloid of Revolution. 

87. Let the axis of the paraboloid be taken as the axis of 
r,. The equation of the parabola being y 2 = 2px, we have, 

dv = 2«pxdx, and xdv = 2*]jx 2 dx. 

Substituting in (1), and integrating from x = to x = a, 

we have, 

2 



II. Moment of Inertia. 

Definitions and Preliminary Principles. 

88. The moment of inertia of a body with respect to an 
axis, is equal to the algebraic sum of the products obtained 
by multiplying the mass of each element of the body by 
the square of its distance from the axis. If we take the 
axis through the centre of gravity of the bodv, and denote 



APPLICATIONS TO MECHANICS AND ASTRONOMY. 17? 

the mass of an element by dm, its distance from the axis 
by x, and the moment of inertia by K, we have, 



■/■ 



E — I x 2 d?n 



(i) 



If we take any parallel axis at a distance d from the 
assumed axis, and denote the moment of inertia with 
respect to it by E', we have (Mechanics, Art. 123). 

K' = K+ md* (2) 



Moment of Inertia of a Straight Line. 

89. Let the axis be taken through the centre of gravity 
of the line and perpendicular to it. Let AB represent the 
line, CD the axis, and E any ele- 
ment. Denote the length of the 
line by 21, its mass by m, the dis- 
tance GE by x, and the length 
of the element by dx. From the 
principle of homogeneity, we have, 







P 


7 








l\ 






R 


E 




G E 






C 





Fig. 19. 



21 : dx 



in 



dm j .\ dm 



dx. 



Substituting in (1), and integrating from x — — I to 
x — + I, we have, 



E= m £; .-. E' = m (¥- + dA 



(3) 



Tnese formulas are entirely independent of the breadth 
of AB in the direction of the axis CD ; they hold good, 
therefore, when the filament is replaced by the rectangle 
EE, the axis being parallel to one of its ends. In this 
case m is the mass of the rectangle ; 21, its length ; and d, 
the distance of the axis from the centre of gravity of the 
rectangle. 



1*8 



DIFEEKENTIAL AND INTEGRAL CALCULUS. 



Moment of Inertia of a Circle. 

. 90. First, let the axis be taken to coincide with one of 
the diameters. Let ACB represent the circle, AB the 
axis, and CD' an element parallel to A 

the axis. Denote 00 by r, OF by x, 
EF by dx, CD' by 2y, and the mass 
of the circle by m. Then, because the 
circle is homogeneous, we have, 



-rr 2 : 2ydx : : m : dm ; 




dm = — -- dx 
irr 2 



2mVr 2 — x 2 



dx. 



rrr' 



Substituting in (1), 



K=^f(r*-x*)lx*dx. 

itr */ 



Reducing by Formulas A and B, integrating by (20), 
and taking the integral between the limits x——r and 
r, = + r, we find, 



K 



= <£; ,X> = m (£ + *) 



(i) 



Secondly, let the axis be taken through the centre and 
perpendicular to the plane of the circle. Let KL be an 
elementary ring, whose radius is x, and 
whose breadth is dx ; then will its area 
be 2<irxdx, and from the principle, that the 
masses are proportional to the volumes, 
we have, 

2mxdo 



irr 9 : 2*xdx 



dm ; .\ dm = 




APPLICATIONS TO MECHANICS AND ASTRONOMY. 179 



Substituting in (1), and integrating from x — to x = r, 
we have, 



mv 



K' = m{^- + d*) 



(5) 



^ , ^ 

Thirdly, to find the moment of inertia of a circulai 
ring with respect to an axis perpendicular to its plane 
Let m denote the mass of the ring, r and r' 
its extreme radii ; 

Then, 7t(r 2 — r /2 ) : %nxdx : : m : dm ; 

2mx 



.-. dm 



ivLlJL . 




Fig. 22. 



r — r 

Substituting in (1) and integrating from 
x = r' to x = r, we have 

2T = ^lO ; ,^ = ro (^ + 4..(6) 

Formulas (5) and (6) are independent of the thickness 
of the plate; hence, they will be true whatever that thick- 
ness may be. Hence, they hold good for a solid and hollow 
cylinder, m being taken to represent the mass of the 
cylinder. 

Moment of Inertia of a Cylinder with respect to an Axis 
perpendicular to the Axis of the Cylinder. 

91. Let the axis be taken through the centre of gravity 
of the cylinder, and let EF be an element perpendiculai 
to the axis of the cylinder. Denote the c 

length of the cylinder by 21, its radius 
by r, its mass by m, the distance of EF 
from the axis by x, and the thickness L 

by dx. Then, as before, we have, pir. 23. 



21 : dx : : m : dm ; 



dm 



dx. 



180 



DIFFERENTIAL AND INTEGRAL CALCULUS. 



The moment of inertia of this element is given by For- 
mula (4) ; but this is the differential of the moment of 
inertia of the entire cylinder ; hence, substituting dm foi 
r», and x for d, in (4), we have, 



:(? + **h 



dK =dr- 



Integrating from — I to + I, we have, 



pi 



Moment of Inertia of a Sphere. 

92. Let the axis pass through the centre of a sphere. 
Let CD' be an elementary segment perpendicular to 
the axis, DC, whose volume is iry 2 dx 
=z<K(r 2 — x 2 )dx ; its mass is found 
from the proportion, 

i 

-in* 3 : ir(r 2 — x 2 )dx \\m\ dm, 

/. dm — -—Ar 2 — x 2 )dx. 
4r 3 

Substituting this value of dm for m in equation (5), and 
making r in that equation equal to y, or to vV 2 — x 2 , we 
have, for the differential of the moment of inertia, 




«HT=f£(r. 



£ 2 ) 2 <;£r. 



Integrating between the limits as = — r, and # = + r, we 
have, 

2mr 2 



K = 



; .-. K' = m(lr 2 + d 2 \ 



APPLICATIONS TO MECHANICS AND ASTRONOMY. 181 

III. Motion op a Material Point. 

General Formulas. 

93. Uniform motion is that in which the moving poini 

posses over equal spaces in equal times ; variable motion 
is that in which the moving point passes over unequal 
spaces in equal times. The velocity of a point is its rate 
of motion. The acceleration due to a force is the rate of 
change that it produces in the velocity of a point. When 
a force acts to increase the velocity, the acceleration is 
positive, when it acts to diminish the velocity, the accelera- 
tion is negative, and conversely. 

Let us denote the mass of a material point by m, its 
velocity at any time t, by v, the space moved over at the 
time t, by s, and the acceleration due to the moving force 
by cp. 

When the motion is uniform, the velocity is constant, and 
its measure is the space passed over in any time divided by 
that time ; but we may, in all cases, regard the motion as 
uniform for an infinitely small time dt ; denoting the 
space described in that time by ds, we have, 

ds 

v = jt (1) 

When the velocity varies uniformly the acceleration is 
constant, and we take for its measure the change of 
velocity in any time divided by that time; but we may, in 
all cases, regard the velocity as varying uniformly for an 
infinitely small time dt ; denoting the change of velocity 
in that time by dv, we have, 



1 82 DIFFERENTIAL AND INTEGRAL CALCULUS. 

In the discussion of motion it is customary to regard the 
time as the independent variable. Differentiating (1), 
under that supposition, we find, 









dv 


d*s t 
~ dt ; 


substituting 


this 


in (2) 

<p = 


, we 

d*s 

dt* 


have, 



(3) 



Equations (1), (2), and (3) are fundamental, and from 
them we may deduce many laws of motion. If we multi- 
ply both members of (3) by m, we have, 

d 2 s „ d*s ,, x 

mv = m W2 ; or, F=m W2 (4) 

In (4), F is the moving force. It will often be conve- 
nient to regard the mass of the material point as the unit 
of mass, in which case, we have, m = 1. 

Uniformly Varied Motion. 

94. Uniformly varied motion is that in which the ve- 
locity increases, or diminishes, uniformly. In the former 
case the motion is uniformly accelerated, in the latter it is 
uniformly retarded, in both the acceleration is constant. 
Denoting the constant value of <p by g, substituting in (3), 
multiplying both mem! jrs by dt, and integrating, we 
have, 

% = gt + 0; ov,v = gt i-0 (5) 

Multiplying again by dt, and integrating, we find, 
s =■ \gt* + Ct + C (6* 



APPLICATIONS TO MECHANICS AND ASTKONOMY. 183 

The constant, 67, is what v becomes when t = ; this is 
called the initial velocity, and may be denoted by v'. The 
constant, C, is what 5 becomes when t = ; this is called 
the initial sjjace, and may be denoted by s'. Making these 
substitutions in (5) and (6), we have, 

v = v' + gt (7) 

8 = *' + v't + igt* . . , . . (8) 

Equations (7) and (8) enable us to discuss all the cir- 
cumstances of uniformly varied motion (see Mechanics, 
Arts. 103 to 108). If ^ represent the force of gravity, and 
v' and s' be each equal to 0, (7) and (8) will express the 
laws of motion when a body falls from rest under the in- 
fluence of gravity, regarded as a constant force. Under 
this supposition they become, 

v=gt (9) 

s = igt* (io) 

That is, the velocities generated are proportional to tne 
times, and the spaces fallen through are proportional to 
the squares of the times. 

Bodies Falling under the Influence of Gravity, regarded as 
Variable. 

95. In accordance with the Newtonian law, the attrac- 
tion exerted by the earth on a body at different distances 
varies inversely as the squares of their distances. Denoting 
the radius of the earth, supposed a sphere, by r, the force 
of gravity at the surface by g, any distance from the cen- 
tre, greater than r, by s, and the force of gravity at that 
distance by <p, we have, 

s* : r* : : g : <p ; .-. <p = ^. 



1&4 DIFFERENTIAL AND INTEGRAL CALCULUS. 

Substituting this value of <p in (3), and at the same time 
making it negative, because it acts in the direction of s 
negative, we have, 

^l = _ffll (11) 

dt* s 3 [ } 

Multiplying by 2ds, and integrating both members, we 
have, 

ds* 2gr 2 „ 9 2gr 2 

If we make v — 0, when s = h, we have, 

aid, ^ = 2^(1-1) (13) 

Equation (13) gives the velocity generated whilst the 
body is falling from the height h to the height s. If we 
make h = 00 , and s = r, (13) becomes, 

v* = 2gr; ,\ v = Vtyr (14) 

In this equation the resistance of the air is not con- 
sidered. 

If in (14) we make g — 32.088 feet, and r — 20923596 
feet, their equatorial values, we find, 

v = 36644 feet, or nearly 7 miles per second. 

Equation (14) enables us to compute the velocity acquired 
by a body in falling from an infinite distance to the sun. 
If we make g = 890.16 feet, and r = 430854.5 miles, which 
are their values corresponding to the sun, we find, 

v = 381 miles per second. 



APPLICATIONS TO MECHANICS AND ASTEONOM*. 185 

If we make li equal to the distance of Neptune, and s 
equal to the sun's radius in (13), we find the velocity that 
a body would acquire in falling from Neptune to the sun, 
under the influence of the sun's attraction. 

To find the time required for a body to fall through any 

ds 
space, substitute -j for v in (13), and solve the results with 

Civ 

respect to dt ; this gives, 
, / h dsVs . / h sds . 

The negative sign is taken because s decreases as t in- 
creases. Reducing (15) by Formula E, and integrating 
by Formula (26), we find, 

' = f/sp^ - ->* -r—- 1 j\+<> < 16 > 

If t = when s = h, we have, 



V 

This, in (16), gives, 



f 2^s X T 



A/^[(hs - ••)* - I /.versin" 1 J + frh\ . . . (1? 



t 

Making s = r in (17), we find 




^[(Ar - r»)* - \ ^versm" 1 ^ + g**] . . , (18) 






Which gives the time required for a body to fall from a 
distance 7i, to the surface of the sun 



186 DIFFERENTIAL AND INTEGRAL CALCULLS. 



Bodies Falling under the Influence of a Force that vaiies as 
the Distance. 

96. It will be shown hereafter, that if an opening were 
made along one of the diameters of the earth, and a body 
permitted to fall through it, the body would be urged 
coward the centre by a force varying as the distance from 
the centre, provided the earth were homogeneous. As- 
suming that principle, and denoting the force at the sur- 
face by g, the radius being r, and the force at the distance, 
t, from the centre by <p, we have, 

r : s : : g : <p ; .-. <p = | & 

Substituting this value of <p in (3), and at the same time 
giving it the minus sign, because it acts in the direction 
of s negative, we have, 

d 2 s _ gs 
di*~ ~~r' 

Multiplying by 2ds, and integrating, 

ds 2 as 2 „ „ a _ 

j-= -*— + G; or, v 2 = C-^-s 3 . 
dt 2 r r 

Making v = 0, when s = r, we have, C =-r 2 ; hence, 

£=?f— ) < 19 > 

If s — 0, we find v = Vgr, which is the maximum velocity. 
It is eqaal to the entire velocity generated by a body fall- 
ing from an infinite distance to the surface of the earth 
divided by V%- If the body pass the centre, s becomes 
n« native, and we find the same values for v at equal dis- 



APPLICATIONS TO MECHANICS AND ASTRONOMY. 18? 

tances from the centre, whether s be positive or negative. 
When s becomes equal to — r, v reduces to 0, and the 
body then falls toward the centre again, and so on con- 
tinually. 

Let us suppose A'C to be the diameter along which the 
body oscillates, and at the time the body starts from A' 
let a second body start from the same 

point and move around the semi-cir- ? ? — 5 — £ 

cumference, A'MC, with a constant y \ j I 

velocity equal to */yr. At any point, n. 5_l \U 
M 9 let this velocity be resolved into J"""^! 

two components, MQ and MN, the 25 

former perpendicular and the latter 
parallel to A'C. The latter component will be equal to 

Vgr. multiplied by co&TMN, or its equal, cosB'MH' ; but 

FT' If 
cosB'MH' — / ; denoting B'H' by s, whence H' M 



vr 2 — s 2 , we have, cos B'MH' 



- zll —j hence, MN= A/ 9 - . (r 2 - s 2 )' 2 ; but this 

is equal to the velocity of the oscillating point when at IT. 
Hence, the velocity of the vibrating point is everywhere 
equal to the parallel component of the velocity of the 
revolving point ; they will, therefore, come together at the 
points A' and C, and the position of the vibrating point 
will always be found by projecting the corresponding posi- 
tion of the revolving point on the path of the former. To 
find the time for a complete vibration from A f to G\ soIvp 
(19) with respect to dt, whence, 



di= -A/-. JZ (20) 



= -l/l ds _ 

V 9 Vr 2 - s 2 



188 DIFFERENTIAL AND INTEGRAL CALCULUS. 

The negative sign is used because Ms a decreasing func- 
tion of s. Integrating (20) between the limits s = r and 
s — — r, we find, 



-y! 



(21) 



This, we shall see hereafter, is the time of vibration of a 
simple pendulum whose length is r. 

The time may be found otherwise, as follows : The space 
passed over by the revolving point is *r ; dividing this by 
the velocity, V#p> we have, 



=Y* 



The species of vibratory motion just discussed is some- 
times called harmonic, being the same as that of a point 
of a vibrating chord or spring. 

Vibration of a Particle of an Elastic Medium. 

97. It is assumed that if a particle of an elastic medium 
be slightly disturbed from its place of rest, and then aban- 
doned, it will be urged back by a force that varies directly 
as the distance of the particle from its position of equilib- 
rium ; on reaching this position, the particle, by virtue of 
its inertia, will pass to the other side, again to be urged 
back, and so on. 

Let us denote the displacement, at any time t, by s, and 
the acceleration due to the restoring force by <p ; then, from 
the law of force, we have, <p == n 2 s, in which n is constant 
for the same medium, under the same circumstances of 
density, pressure, etc. Substituting for <p its value from 
Equation (3), and prefixing the negative sign, because it 



APPLICATIONS TO MECHANICS AND ASTRONOMY. 189 

acts in a direction contrary to that in which s is estimated, 
we have, 

-&=*» ^ 

Multiplying by 2ds, and integrating, we have, 

-^ =n>*« 4- = -v* (23) 

The velocity is when the particle is at the greatest 
distance from the position of equilibrium ; denoting this 
value of s by a, we have. 

n 2 a 2 + C=0; .-. G= — n 2 a 2 , 

which, in (23), gives, 

f£_ = n *(a* - s 2 ) ; or, ndt = -=== (24) 

at' ^/ a 2 _ s z 

Integrating (24), we have. 

nt + G = sin" 1 - (25) 

a v ' 

Taking the sines of both members, and reducing, we have, 

s = asm(nt + C) (26) 

Tf we make t = 0, when s = 0, we have C = 0, ana Equa- 
tion (26) becomes, 

s = a $m(nt) (27) 

If (25) be taken between the limits — a and -f a, we find 
for the time of a single vibration, denoted by \T, 



J 90 DIFFERENTIAL AND INTEGRAL CALCULUS. 

Substituting this in (27), we have, finally, 

s = asml — t ) ( 38 ) 

in which T is the time of a double vibration. 
Solving (24), we have, 
ds 



— - = v = n 



dt 



Va* - s z (29) 



Substituting for 5, in (Id), its value from (28), we find, 
v = nA/a* - a* sin* (^t} = nai/l - sin*^ /) 



Whence, we have, 

'2<7r 



v = nacos(—t\ (30) 



This equation is used in discussing the laws of light, 
and in many other cases. 

Curvilinear Motion of a Point. 

98. A point cannot move in a curve except under the 
action of an incessant force, whose direction is inclined to 
the direction of the motion. This force is called the de- 
flecting force, and can be resolved into two components, 
one in the direction of the motion, and the other at right 
angles to it. The former acts simply to increase or di 
minish the velocity, and is called the tangential force ; the 
latter acts to turn the point from its rectilinear direc- 
tion, and being directed toward the centre of curvature 
is called the centripetal force. The resistance offered by 
the point to the centripetal force, in consequence of its 



APPLICATIONS TO MECHANICS AND ASTRONOMY. 191 

inertia, is equal and directly opposed to the centripetal 
force, and is called the centrifugal force. 

To find expressions for the tangential and centripetal 
forces, let the acceleration due to the deflecting force in the 
direction of the axis of x, at any 
time t, be denoted by X, and the 
acceleration in the direction of 
the axis of y by Y. Let these be 
resolved into components acting 
tangentially and normally. The 
algebraic sum of the tangential 
components is the tangential accel- 
eration denoted by T, and- the 
algebraic sum of the normal components is the centripetal 
acceleration denoted by N. Assuming the notation of the 
figure, we have, 

T — Xcos2 + Ysind ; 
N = Xsind - Ycos&. 




Fig. 26. 



But from Articles (93) 


and (5), we 


have, 




X = 


d 2 x 
dt 2 '' 


' dt 2 ' 


and, 










COS0 : 


dx . , 


dy 



Substituting in the preceding equations, and reducing oii 
the supposition that t is the independent variable, and b^ 
means of Formula (3), Art. (38), we have, 



T = 



d 2 x dx d 2 y dy 
U 2 'Ts + W fs 



d (dx* + dy 2 ) d(ds* ) _ d*s 
dt 2 . 2ds ~ rI1*2ds "~ dt 2 



(31) 



192 DIFFERENTIAL AND INTEGRAL CALCULUS. 

._ __ d*xdy — d*ydx 

~ dI*Tds 

ds 2 d f ydx — d 2 xdy _ v % 
~ ~ dt* ' ~^ 5" ~ ~ R 

But the centripetal acceleration is equal and directly op 
posed to the centrifugal acceleration. Denoting the lattei 
by f, we have, 

f= v i < 32 > 

From (31) we see that the tangential force is in depend 
ent of the centripetal force, and from (32) we see that the 
acceleration due to the centrifugal force at any point of the 
trajectory, is equal to the square of the velocity divided bv 
the radius of curvature at that point. 



Velocity of a Point rolling down a Curve in a Vertical Plana, 

99. Let a point roll down a curve, situated in a vertical 
plane, under the influence of gravity regarded as constant. 
Let the origin of co-ordinates be taken at the starting 
point, and let distances downward be positive. At any 
point of the curve, whose ordinate is y, the force of gravity 
being denoted by g, we have, for the tangential component 

dy 

of gravity, #sin<3, or, g -j- ; placing this equal to its value, 

equation (31), we have, 

dy d 2 s , ds . d 2 s 

Integrating and reducing, remembering that the constant 
is under the particular hypothesis, we have, 

1 ds2 » o rr~ 
& = o T77i ; or > v = tyy- •'• v ~ vtyy (1) 



2 dV 



APPLICATIONS TO MECHANICS AND ASTRONOMY. 193 

Tne second member of (1) is the velocity due to the 
height y. Hence, the velocity generated by a body rolling 
down a curve, gravity being constant, is equal to that gen- 
erated by falling freely through the same vertical height. 

This principle is true so long as g is constant. But g 
may be regarded as constant from element to element, no 
matter what may be the law of variation. Hence, if a body 
fall toward the sun, or earth, on a spiral line, the ultimate 
velocity will be the same as though it had fallen on a right 
line toward the centre of the attracting body. The direc- 
tion of the motion, however, is not the same, for in the 
former case it is tangential to the trajectory pursued, and 
in the latter case it is normal to the attracting body. 

The Simple Pendulum. 

100. A simple pendulum is a material point suspended 
from a horizontal axis, by a line without weight, and free 
to vibrate about that axis. 

Let ABC be the arc through which the vibration takes 
place, and denote its radius by I. The angle CD A is the 
amplitude of vibration ; half this angle, 
ADB, denoted by a, is the angle of 
deviation; and I is the length of the 
pendulum. If the point start from 
rest, at A, it will, on reaching any 
point, H, of its path, have a velocity, v, 
due to the height EK, denoted by y. B 

Hence, 

v = V%gy. 

If we denote the angle HDB by d, we have DK — /cosfl ; 
we also have DE=lcosa ; and since y is equal to DE—DE, 
we have, 

y ~ t T {COS0 — COSa), 



194 DIFFERENTIAL AND INTEGRAL CALCULUS. 

which, being substituted in the preceding formula, 

gives, 

v = V^?(cosd — COSa). 



Again, denoting the angular velocity at the time t by #, 

id rememberi 
the pendulum, 



(U 
and remembering that <p = — , we have, for the velocity of 



id* 
v = lx, = l Jt . 

Equating the two values of v, and reducing, we have, 



dt 



VI 



Vcos4 



COSa 



Developing cosfl and cosa by McLaurin's Formula, we have, 



cosfl = 1 — — + — — etc. 

Z /c4 



«2 a * 
COSa = 1 - — + — - etc. 



If we suppose the amplitude to be small, we may neglect 
all the terms after the second ; doing so, and substituting 
in (31), we have, 



y g V* 2 -& 2 



Integrating between the limits 6 = — a, and <3 = + a, we 
Gnd. 



VS 



which is the formula for the time of vibration of a simpU 
pendulum. 



APPLICATIONS TO MECHANICS AND ASTKON T OMV. 195 



Attraction of Homogeneous Spheres. 

101. The Newtonian Law of universal gravitation may 
be expressed as follows, viz. : every particle of matter 
it tracts every other particle, with a force that varies 
directly as the mass of the attracting particle, and inversely 
as the square of the distance between the particles. To 
apply this law to the case of homogeneous spheres, let us 
first consider the action of a spherical shell, of infinitesimal 
thickness, on a material point within it. 

Let D be the material point, APB a great circle through 
it, and let the diameter ADB be taken as the axis of x. 
If the circle be revolved 
about AB, its circum- 
ference will generate a 
spherical shell, and any 
element of the circumfer- 
ence, as P, will generate 
an elementary zone whose 
altitude is dx. For any 
point of this zone, as P, 
there is another point, P f 
symmetrical with it, and the resultant action of these points 
on D is directed along AB, and is equal to the sum of the 
forces into the cosine of the angle BDP. Hence, the re- 
sultant attraction of the entire zone is directed along AB, 
and is equal to the sum of the attractions of all its particles 
into the cosine of the angle BDP. To find an expression 
for this resultant, denote CP by r, CD by a, DP bv z 5 
and CEhj x. Because the shell is infinitesimal in thick- 
ness and homogeneous, the mass of the zone, is to the mass 
of the shell, as dx, the altitude of the zone, is to %r, the 
altitude of the shell. Hence, if the mass of the shell be 




Fig. 28. 



L96 



DIFFERENTIAL AND INTEGRAL CALCULUS. 



denoted by m, the mass of the zone will be denoted b) 

— — . If the force exerted by the unit of mass at the unit 

of distance be taken as the unit of force, the attraction 

exerted by all the particles of the zone on the point D will 

tizclx 1 
be equal to -^— . — , and the resultant action on D, in the 

<C7 Z 

direction of AB, denoted by df, will be given by the equa- 
tion, 

df=~.\,.^EDP (1) 



From the triangle, PCD, we have, 
DP 2 = z 2 = r 2 + a 2 — 2ax = r 2 - a' 
and from the triangle, EDP, we have, 

cosEDP = ^-^. 



2a(x — «), 



Substituting in (1), remembering that dx equals d(x — a) 
we have, 



df = 



(x — a) d(x — a) 



[r 2 - a 2 — 2a(x - a)J' 



(2) 



Regarding (x — a) as a single variable, reducing by For 
mula A, and integrating by Formula (29), we find, 






x — a 



[r 2 -a 2 - 2a{x-a)Y 



a 2 \r' 



r 2 - a 2 ) 

- a 2 - 2a(x -a)]$S 



Taking the integral from x = — r to x ~ + r, we have, 



APPLICATIONS TO MECHANICS ANO ASTRONOMY. 1^7 

Hence, the effect of the attraction of the shell on any 
point within it is null. If a sphere be described about C 
as a centre with a radius equal to a, we may ca'l that part 
which lies between it and the surface of the given spher*"' 
the exterior shell, and the sphere itself may be called the 
nucleus. From w r hat precedes, we infer that any point 
within a homogeneous sphere is acted on by the sphere 
precisely as though the exterior shell did not exist. Hence, 
a point at the centre of a sphere is not affected by the 
attraction of the sphere. 

If the point D' be taken without the shell, we have, 

jyp 2 = z 2 = r 2 4- a 2 4- 2ax = r 2 - a" 4- 2a(x 4- a), 
and from the triangle, ED'P, we have, 

cosBD'P = ?±£. 
z 

Substituting in (1), we have, 

df= £ (* + ")*(*+*) 3 (3 ) 

% r [v - a? + 2a(x + a)]' 2 

Reducing and integrating as before, we find, 



f _ m ( x 4- a 

2r ( a[r 2 -a 2 + 2a(x + a)] 2 



--a' 



a 2 [r 2 -a 2 + 2a(x 4- a) 



4- C 



Taking the integral between the limits x = — ; and 
r = t r, we have, when a > r 

r=~, (4) 



L98 differential and integral calculus. 

But m is the mass of the shell, and a is the distance of D 
from the centre; hence, the shell attracts a particle with- 
out it as though its entire mass were concentrated at its 
centre. 

A homogeneous sphere may be regarded as made up of 
spherical shells; hence, a sphere attracts any point without 
it, as though the mass of the sphere were concentrated at its 
centre. The same is true of a sphere made up of homoge- 
neous strata, which vary in density in passing from the 
surface to the centre. It is to be inferred that two homo- 
geneous spheres, or two spheres made up of homogeneous 
strata, attract each other as though both were concentrated 
at their centres. 

If an opening were made from the surface to the centrt: 
of the earth, supposed homogeneous, and a body were to 
move along it under the earth's attraction, it would every- 
where be urged on by a force varying directly as the dis- 
tance of the body from the centre. For, denote the dis- 
tance of the body from the centre at any instant by x. 
The body will only be acted on by the nucleus whose radius 
is x. If we take r to represent the radius of the earth, 
and remember that the masses are proportional to these 
volumes, we have, 

4 4 

M : m : : ^Tr 3 : ^^x 3 : : r 3 : x*. 

O 

Denoting the force of attraction at the surface by g, and 
the force of attraction at the point whose distance from 
the centre is x by/, we have, from the Newtonian law. 

v 3 x 3 o 

9-f ■■■■^■■^ ■■■?■■ *; .■•/=*» 

Hence, the proposition is proved. (See Art. 96.) 



APPLICATIONS TO MECHANICS AND ASTRONOMY. L99 



Orbital Motion. 

102. If a moving point, P, be continually acted on by a 
JL fleeting force directed toward a fixed centre, it will de- 
scribe a line or path, called an orbit. If the moving point 
be undisturbed by the action of any other force, the orbit 
will lie in a plane passing through the fixed centre and an 
element of the curve. Let 
this plane be taken as the 
co-ordinate plane, let the 
fixed point be the origin, 
and let the orbit be repre- 
sented by APB. 

Denote the acceleration 
due to the deflecting force 
at any time t by/, its in- 




Fig. 29. 



clination to the axis by <p, and its components in the direc- 
tions of the co-ordinate axes by X and Y. 
From the figure, we have, 



X — — fcosv, and T = — /sin<p 



(i) 



If we regard t as the independent variable, dx and dy will 
both be variable, and from Equation (3), Art 93, we have. 



_ d 2 x , Tr d 2 y 



U we denote the co-ordinates of P by x and y, and its 
radiu3 vector FP by r, we have, from the figure. 



x II 

costp = -, sincp = -; 
r r 



and 



x 2 + y 2 = r 2 ; :. xdx + ydy — rdr. 



200 DIFFERENTIAL AND INTEGRAL CALCULUS* 

Substituting in (1), we have, 

d 2 % /.« , d 2 y ,y /ox 

If* = ~f? and It* = ~ f r < 2 > 

Multiplying the first of Equations (2) by y, the second bj 
x, and subtracting the former from the latter, we have, 

xd 2 y iicPx . d (xdti — ydx) 

-J-- 9 -W = 0> or » M > = (3) 

Multiplying the first by dx, the second by dy, adding tht 
resulting equations and reducing, we have, 

dxd 2 x dyd 2 y Jt xdx -f ydy ., /JX 

Sr + dW = -1 r = S* W 

Integrating (3) and (4), Art. 77, we have, 



and 



dy dx „ ,„. 



+ ^/tfr = C7" (6) 



or, 

Equations (5) and (6) make known the circumstances 
of motion when the value of / is given. It is found con- 
venient to transform them to a system of polar co-ordinates, 
whose pole is F, and whose initial line is FX. The for- 
mulas for transformation are 

x — rcoep, and y = rsinp (7) 

Hence, by differentiating, we have, 

dy dr . dq> ,„ 

di = dt sm,f + rao ^dt (8 > 

and 

dx dr . dy /M 

# = !i eos > ~ rsin *Tt (9 > 



APPLICATIONS TO MECHANICS AND ASTHOifOMY. 201 

We also, have, Art. (53), 

ds 2 = r 2 d(p 2 + dr 2 (10) 

Substituting in (5), and reducing by the relations, 

t . x 2 + y 2 

x&mcp — y 30-sp = 0, and zcoscp + 2/sintp = — r t 

we have, 



From equation (6), we have, 



dr 2 +r 2 dq> 2 
dt 2 



+ %ffdr = G" (12) 



But, from (11), we have, 

r 2 d(D -, m r*d(p 2 

dt = —^-, or, <#« = -^-, 

and this, in (12), gives, 

° s \^ + ^\ + 2 f fclr = " (13) 

Equations (11) and (13) are the equations required. 
Multiplying (11) by dt, and integrating, we have. 



I r 2 dy = 



Ct + C" (14) 



But, by Art. 53, r 2 dq> is twice the elementary area swept 
over by the radius vector; hence, the first member of (14) 
is an expression for twice the area swept over by the radius 
vector up to the time t. If we suppose the area to be 
reckoned from the initial line FX, and at the same time 



202 DIFFERENTIAL AND INTEGRAL CALCULUS. 

suppose t — 0, we have C" = 0. Substituting this in (14), 
and in the resulting equation making t — 1, we find 
C equal to twice the area swept over in the first unit of 
time ; denoting the area described in the unit of time bv 
I, we have, G = 2 A, and consequently, 



/ 



r*dcp = 2At (15) 

From Equation (15) we infer that the areas described by 
che radius vector are proportional to the times of descrip- 
tion, and this without reference to the nature of the de- 
flecting force. 

To find the equation of the orbit, let us assume as h 
particular case, the Newtonian law of universal gravita- 
tion. Denoting the force exerted by the central body at a 
unit's distance by k, we have, for the attraction at the 
distance r, 

f=£ < 1C ) 

Making r = -, whence dr = r, and substituting for L 

° u u z ° 

ind/ their values in (13), we have, 



<>r, 



du 2 . „ 2k 



dv* +U *=iA* U + C ^) 



,v i . dii 



To find the value of C , let — - = 0, when r — r\ or 

dii dv 

u = u'. But when — - = 0, we also have -— = 0, that is, 
dq> a$ 



APPLICATIONS TO MECHANICS AND ASTRONOMY. 203 

ihe radius vector is perpendicular to the arc. If we denote 
the corresponding velocity by v', we have, 



A VT A A* > 2 > 2 

A = — , or, &A* = v r ; 



and, consequently, 



2k 2h 



■u = 



. A 2 ,2 ,3' 

4cA v r 



Making these substitutions in (18), we find, 

~y X lifC 

6 —zr* 



Denoting this value of C Y by S, and solving Equation (18), 
we have, 

du 2 ( > M \ o h * f & V 

^ = S -[ U2 -^ U ) = S + 16A^-{ U -IT 2 ) 

= p 2 - (u -q) 2 . 

Solving with reference to <p, and taking the negative sign 
of the radical, we have, 

, du du — q) 

d(p = = ±!= - (19; 

Vp 2 — (u - q) 2 Vp 2 — (u - q) 2 

Integrating, we have, 

<p - a = COS" 1 ^^ (20) 

■ p 

In which — a is an arbitrary constant. If we suppose 
p -= 0, when v — v' t r = r', and u = u', the corresponding 
value of a is the angular distance from the initial line to 
the radius vector whose direction is perpendicular to the 
element of the curve. 



204 DIFFERENTIAL AND INTEGRAL CALCULUS. 

Taking the cosines of both members of (20), we have, 

i = cos(<p — a), or, u = q + i?cos(<p — a) 

and finally replacing u by its value, and solving, we have. 
1 4^2 



q +pcos(y — a) k+ V 16SA 4 + k 2 X cos(<p — a) 

(21) 

Equation (21) is the polar equation of a conic section. 
Hence, the orbit of a particle about a central attracting 
body is one of the conic sections. If the orbit is a closed 
curve, it must be an ellipse. This corresponds to the case 
of a planet revolving about the sun, or to that of a satellite 
revolving about its primary. 

Law of Force. 

103. If the orbit of a revolving particle be an ellipse, it 
must be attracted to the focal point by a force that 
varies inversely as the square of the distance. 

The polar equation of an ellipse is 

r= "P-*') m 

1 + ecos(cp — a) 

in which a is the semi-transverse axis, e is the eccentricity, 
a is the angular distance from the fixed line to the radius 
vector drawn to the nearest vertex, or the longitude of tb» 
perihelion in the case of a planetary orbit. The angle, 
r> — a, is the angle that is called in astronomy the true 
anomaly. 

Putting r = -, in (1), we get, 
u 

* _ - atl-e') (2) 






APPLICATIONS TO MECHANICS AND ASTKONOMY. 205 

Differentiating twice, with respect to <p, we find, 

du _ esin((p — a) 

a%~ ~ ~a{l-e 2 ) ( ■ 

ftnd, 

d 2 u _ ecos((p — a) 

d^~ ~ a(l - e 2 ) ( ' 

Adding (2) and (4), we have, 

d 2 u 1 /B , X 

a? + u = 7tr^) ( 5 > 

Resuming equation (13) of the last article, replacing r by 

its value, -, and dr by its value, r , differentiating and 

reducing, we have, 

HS-W = ° w 

Combining (5) and (6), and solving with respect to /, wp 
have, 

C 2 u 2 C 2 1_ 
' ~ «(i _ 6 2y or / - fl (i _ e 2) • r % I 7 ) 

Hence, / varies inversely as the square of r, which was to 
be shown. 



Note on the Methods of the Calculus. 

104. All the rules and principles of the Calculus, in the 
present treatise, have been deduced in accordance with the 
method of infinitesimals, as explained in Articles 6 and 7. 
They might also have been deduced by the method of limits. 
It remains to be shown that the results obtained by these 
two methods are always identically the same, 



*06 DIFFERENTIAL AND INTEGRAL CALCULUS. 

To explain the method of limits, let us denote any func- 
tion of # by y ; that is. let us assume the equation, 

v=a*) a) 

' I we increase x by a variable increment h, and denote the 
>orresponding value of y by y', we have, 

y'=f(x + h) (2) 

It is shown in Courtenay's Calculus, Article 4, that so long 
as x retains its general value, the new state of the function 
nan be expressed by the formula, 

y' =f( x + h) =f(x) + Ah + Bh* + Ch* + etc. . . (3) 

in which A, B, C, etc., depend on x, but are independent 
of h. If we subtract (1) from (3), member from member. 
we have, 

y' - y = Ah + Bh* + (etc.) h* (4) 

Dividing both members of (4) by h, we have, 

V -^~- = A + Bh + (etc.) h* . • '. . . (5) 

The first member of (5) is a symbol to express the ratio 
of the increment of the variable, to the corresponding incre- 
ment of the function, and the second member is the value 
of that ratio. 

It is shown in Algebra, that in an expression like the 
second member of equation (5), it is always possible, to 
give to h a value small enough to make the first term 
numerically greater than the algebraic sum of all the 
others. If we assign such a value to h, and then suppose 
h to go on diminishing, the second member will conrin- 
nally approach A, and when h becomes 0, the second mera 



APPLICATIONS TO MECHANICS AND ASTRONOMY. 207 

o^f will reduce to A. Hence, A is a quantity toward 

v' — y 

which the ratio, - — 7-—, approaches as h is diminished, but 

beyond which it cannot pass ; it is therefore the limit of 
that ratio. This limit is the differential coefficient of y, 
and, as may easily be seen, it is entirely independent of dx. 
The product of this by the differential of the variable ia 
Lite differential of y. 

We may, therefore, enunciate the method of limits as 
follows: viz., Give to the independent variable a variable 
increment, and find the corresponding state of the function ; 
from this subtract the primitive state, and divide the dif- 
ference by the variable increment ; then pass to the limit 
of the quotient by making the increment of the variable 
equal to ; the result is the differential coefficient of the. 
function ; if this be multiplied by the differential of the 
variable, the product is the differential of the function. 

In the case assumed, we have, by this method, 

J £=A; .:dy=Adx (6) 

If we make h — dx, in Equation (4), dx being infinitely 
small, the first member will be the differential of y, and 
all the terms of the second member after the first may be 
neglected. Hence, by the method of infinitesimals, we 
have, 

dy = Adx ; .*. -f = A. 

ax 

This result is the same as that obtained by the method of 
limits. But, by hypothesis, y represents any function of 
x ; hence, in all cases, the differential coefficient is iden- 
tically the same whether found by the method of limits or 
by the method of infinitesimals. 



tf)8 DIFFERENTIAL AND INTEGRAL CALCULUS. 

Every function, regarded as a primitive, is connectet 
with some other function, regarded as a derivative, by the 
law of differentiation. This derivative is the differential 
coefficient of the primitive. The object of the differential 
calculus is to find the derivative from its primitive; the 
object of the integral calculus is to find the primitive from 
its derivative; every application of the calculus depends 
on one of these processes, or on some discussion growing 
out of one, or the other. Now, because the primitive and 
the derivative are independent of the differentials of both 
function and variable, the relation between them is, of 
necessity, independent of the methods employed in estab- 
lishing that relation. But it has been shown that the 
same relation is found between these functions, whethei 
we employ the method of infinitesimals or the method of 
limits. Hence, these methods, and the results obtained b) 
them, are in all cases logically identical. 






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inductive in method, lucid in style, orderly in arrangement, 
and clear and comprehensive in treatment. Sufficiently 
elementary for the lower grades of high school classes and 
complete enough for all secondary schools. 



Copies of the above books will be sent prepaid to any address, on receipt 
of the price , by the Publishers : 

American Book Company 

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(80) 



Eclectic English Classics for Schools 

This series is intended to provide selected gems of English Literature 
for school use at the least possible price. The texts have been carefully- 
edited, and are accompanied by adequate explanatory notes. They are 
well printed from new, clear type, and are uniformly bound in boards. 
The series now includes the following works : 

Arnold's (Matthew) Sohrab and Rustum . . . .20 cents 

Burke's Conciliation with the American Colonies . . .20 cerits 

Carlyle's Essay on Robert Burns 20 cents 

Coleridge's Rime of the Ancient Mariner . . . .20 cents 

Defoe's History of the Plague in London . . . . 40 cents 

DeQuincey's Revolt of the Tartars ..... 20 cents 

Emer sons American Scholar, Self-Reliance, and Compensation 20 cents 

Franklin's Autobiography . . . . . . . 35 cents 

George Eliot's Silas Marner ....... 30 cents 

Goldsmith's Vicar of Wakefield ...... 35 cents 

Irving's Sketch Book — Selections . • 20 cents 

Tales of a Traveler . . . . . . . 50 cents 

Macaulay's Second Essay on Chatham 20 cents 

Essay on Milton ........ 20 cents 

Essay on Addison .. ^ ..... 20 cents 

Life of Samuel Johnson ....... 20 cents 

Milton's L' Allegro, II Penseroso, Comus, and Lycidas . . 20 cents 
Paradise Lost — Books I. and II. . . . . .20 cents 

Pope's Homer's Iliad, Books I., VI., XXII. and XXIV. . 20 cents 

Scott's Ivanhoe 50 cents 

Marmion . . . . . . . . . 40 cents 

Lady of the Lake 30 cents 

The Abbot 60 cents 

Woodstock 60 cents 

Shakespeare's Julius Caesar 20 cents 

Twelfth Night 20 cents 

Merchant of Venice 20 cents 

Midsummer-Night's Dream ...... 20 cents 

As You Like It 20 cents 

Macbeth 20 cents 

Hamlet 25 cents 

Sir Roger de Coverley Papers (The Spectator) . . .20 cents 

Southey's Life of Nelson 40 cents 

Tennyson's Princess 20 cents 

Webster's Bunker Hill Orations 20 cents 



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Study of American Literature H 



BY > 

BRANDER MATTHEWS 

Professor of Literature in Colurafeia University' 4 

Y . r - h *» • * 

Cloth, J2mo, 256 pages' -V v - • - 'P4ce; $1.00 






A text-book of literature on an original plan, and conforming with 
the best methods of teaching. J* % * 

Admirably designed^©* guide, to supplement, '.and to stimulate the 
student's reading of American authors. 

Illustrated with a fine collection of facsimile manuscripts, portraits 
of authors, and views of tliejr homes and birthplaces. 

Bright, clear, and fascinating, it is itself a literary work of high rank. 

The book consists mostly of delightfully readable and yet compre- 
hensive little biographies of the "fifteen greatest and most representative 
American writers. Each of the sketches contains a critical estimate of 
the author and his works, whichjs the more valuable coming, as it does, 
from one who is 'himself a master^ The work is rounded out by four 
general chapters which take up other prominerrt authors and discuss the 
history and conditions of our literature as a whole ; and there is at the 
end of the book a complete chronology of the best American literature 
from the beginning down to r8o,6. 

Each ,of the fifteen biographical sketches is illustrated by a fine 

'portrait of its subject and views of his birthplace or residence and in 

some cases ot befth. They are also accompanied by each author's 

facsimile manuscript covering one or two' pages. The .book contains 

excellent portraits of many other authors famous jn American literature. 



Copies -of Brander Afattheios' Introduction to the Study of A7nerican 
Literature jzvill 3e sent* .prepaid to any address', on receipt of the price, 
by the Publishers J. y 

' « American .Book Company 

New York" : * ♦. ^Cincinnati ^ ♦ Chicago 



•* 



- -i ■ 



^Fisher's Brief History of the Nations 

AND OF THEIR PROGRESS IN CIVILIZATION- 
By GEORGE PARK FISHER, LL.D. 

Professor in Yale University 

Cloth,«12mo, 613 pages, with numerous Illustrations, Maps, Tables, and 
Reproductions of Bas-reliefs, Portraits, and Paintings. Price, $1.50 



This is an entirely new work written expressly to meet 
the demand for a compact and acceptable text-book oil 
General History for high schools, academies, and private 
schools. Some of the distinctive qualities which will com- 
mend this book to teachers and students are as follows: 

It narrates in fresh, vigorous, and attractive, style the 
most important facts of history in their due order and 
connection. 

It explains the nature of historical evidence, and records 
only well established judgments respecting persons and 
events. 

It delineates the progress of peoples and nations in 
civilization as well as the rise and succession of dynasties. 

It connects, in a single chain of narration, events related 
to each other in the contemporary history of different 
nations and countries. 

It gives special prominence to the history of the 
Mediaeval and Modern Periods, — the eras of greatest 
import to modern students. 

It is written from the standpoint^ of the present, and 
incorporates the latest discoveries of historical explorers-* 
and writers. 

It is illustrated by .numerous colored maps, genealog- 
ical tables, and artistic reproductions of architecture^ 
sculpture, painting, and portraits of celebrated meti, 
representing every periool of tt>e world's history. 



Copies of Fisher s Brief History of the Nations will be sent prepaid to 
any address, on receipt of the price, by the Publishers ; 



American Book Company 



New York ♦ Cincinnati ♦ Chicago 

(43) 






Handbook of Greek and Roman History 

BY 

GEORGES CASTEGNIER, B.S., B.L. 



Flexible Cloth, 12mo, 110 pages. - • Price, 50 cents 



The purpose of this little handbook is to assist the 
student of Greek and Roman History in reviewing subjects 
already studied in the regular text-books and in preparing 
for examinations. It will also be found useful for general 
readers who wish to refresh their minds in regard to the 
leading persons and salient facts of ancient history. 

It is in two parts, one devoted to Greek, and the other 
to Roman history. The names and titles have been 
selected with rare skill, and represent the whole range of 
classical history. They are arranged alphabetically, and 
are printed in full-face type, making them easy to find. 
The treatment of each is concise and gives just the in- 
formation in regard to the important persons, places, and 
events of classical history which every scholar ought to 
know and remember, or have at ready command. 

Its convenient form and systematic arrangement 
especially adapt it for use as an accessory and reference 
manual for students, or as a brief classical cyclopedia for 
general readers. 



Copies of Castegniers Handbook of Greek and Roman History will be 
sent prepaid to any address \ on receipt of the price, by the Publishers: 

American Book Company 

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(44) 



Halleck's Psychology and 
Psychic Culture 

By REUBEN POST HALLECK, M.A. (Yale) 
Cloth, 12mo, 368 pages. Illustrated .... Price, $1.25 



This new text-book in -Psychology and Psychic Culture 
is suitable for use in High School, Academy and College 
classes, being simple and elementary enough for beginners 
and at the same time complete and comprehensive enough 
for advanced classes in the study. It is also well suited 
for private students and general readers, the subjects being 
treated in such an attractive manner and relieved by so 
many apt illustrations and examples as to fix the attention 
and deeply impress the mind. 

The work includes a full statement and clear exposition 
of the coordinate branches of the study — physiological and 
introspective psychology. The physical basis of Psychol- 
ogy is fully recognized. Special attention is given to the 
cultivation of the mental faculties, making the work 
practically useful for self-improvement. The treatment 
throughout is singularly clear and plain and in harmony 
with its aims and purpose. 

" Halleck's Psychology pleases me very much. It is short, clear, 
interesting;, and full of common sense and originality of illustration. 
I can sincerely recommend it." 

WILLIAM JAMES, 
Professor of Psychology, Harvard University. 



Copies of Hallectis Psychology will be sent prepaid to any address on 
receipt of the price by the Publishers : 

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(42) 




Mythology 



Guerber's Myths of Greece and Rome 

Cloth, l2mo, 428 pages. Illustrated .... $1.50 

Guerber's Myths of Northern Lands 

Cloth, i2mo, 319 pages. Illustrated .... $1.50 

Guerber's Legends of the Middle Ages 

Cloth, i2mo, 340 pages. Illustrated .... $1.50 

By H. A. Guerber, Lecturer on Mythology. 

These companion volumes present a complete outline of Ancient, 
and Mediaeval Mythology, narrated with special reference to Literature 
and Art. They are uniformly bound in cloth, and are richly illustrated 
with beautiful reproductions of masterpieces of ancient and modern 
painting and sculpture. 

While primarily designed as manuals for the use of classes in schools 
where Mythology is made a regular subject of study and for collateral 
and supplementary reading in classes studying literature or criticism, 
they are equally well suited for private students and for home reading. 
For this purpose the myths are told in a clear and charming style and in 
a connected narrative without unnecessary digressions. To show the 
wonderful influence of these ancient myths in literature, numerous and 
appropriate quotations from the poetical writings of all ages, from 
Hesiod's " Works and Days " to Tennyson's " CEnone," have been in- 
cluded in the text in connection with the description of the different 
myths and legends. 

Maps, complete glossaries and indexes adapt the manuals for conven- 
ient use in schools, libraries or art galleries. 



Copies of the above books will be sent prepaid to any address, on receipt of 
the price, by the Publishers: 

American Book Company 

New York ♦ Cincinnati ♦ Chicago 

(37) 



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